Difference between revisions of "Metric Spaces"

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In [[mathematics]], a '''metric space''' is a [[Set (mathematics)|set]] together with a [[Metric (mathematics)#Definition|metric]] on the set. The metric is a [[function (mathematics)|function]] that defines a concept of ''distance'' between any two [[Element (mathematics)|members]] of the set, which are usually called [[point (geometry)|points]].  The metric satisfies a few simple properties. Informally:
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In mathematics, a '''metric space''' is a set together with a metric on the set. The metric is a function that defines a concept of ''distance'' between any two members of the set, which are usually called points.  The metric satisfies a few simple properties. Informally:
 
* the distance from <math>A</math> to <math>B</math> is zero if and only if <math>A</math> and <math>B</math> are the same point,  
 
* the distance from <math>A</math> to <math>B</math> is zero if and only if <math>A</math> and <math>B</math> are the same point,  
 
* the distance between two distinct points is positive,   
 
* the distance between two distinct points is positive,   
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* the distance from <math>A</math> to <math>B</math> is less than or equal to the distance from <math>A</math> to <math>B</math> via any third point <math>C</math>.
 
* the distance from <math>A</math> to <math>B</math> is less than or equal to the distance from <math>A</math> to <math>B</math> via any third point <math>C</math>.
  
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A metric on a space induces [[topological properties]] like [[open set|open]] and [[closed set]]s, which lead to the study of more abstract [[topological space]]s.
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A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces.
  
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The most familiar metric space is [[3-dimensional Euclidean space]]. In fact, a "metric" is the generalization of the [[Euclidean metric]] arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the [[Straight line|straight]] [[line segment]] connecting them. Other metric spaces occur for example in [[elliptic geometry]] and [[hyperbolic geometry]], where distance on a [[sphere]] measured by angle is a metric, and the [[hyperboloid model]] of hyperbolic geometry is used by [[special relativity]] as a metric space of [[velocities]]. Some of non-geometric metric spaces include spaces of finite strings ([[finite sequence]]s of symbols from a predefined alphabet) equipped with e.g. a [[Hamming distance|Hamming]]'s or [[Levenshtein distance]], a space of subsets of any metric space equipped with [[Hausdorff distance]], a space of [[Real-valued function|real functions]] [[integral|integrable]] on a unit [[interval (mathematics)|interval]] with an integral metric <math>d(f,g)=\int_{x=0}^{x=1}\left\vert f(x)-g(x)\right\vert\,dx</math> or probabilistic spaces on any chosen metric space equipped with [[Wasserstein metric]]. See also the section {{Section link||Examples of metric spaces}}.
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The most familiar metric space is 3-dimensional Euclidean space. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities. Some of non-geometric metric spaces include spaces of finite strings (finite sequences of symbols from a predefined alphabet) equipped with e.g. a Hamming's or Levenshtein distance, a space of subsets of any metric space equipped with Hausdorff distance, a space of real functions integrable on a unit interval with an integral metric <math>d(f,g)=\int_{x=0}^{x=1}\left\vert f(x)-g(x)\right\vert\,dx</math> or probabilistic spaces on any chosen metric space equipped with Wasserstein metric. See also the section Examples of metric spaces.
  
 
== Definition ==
 
== Definition ==
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A '''metric space''' is an [[ordered pair]] <math>(M,d)</math> where <math>M</math> is a set and <math>d</math> is a [[metric (mathematics)|metric]] on <math>M</math>, i.e., a [[Function (mathematics)|function]]
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A '''metric space''' is an ordered pair <math>(M,d)</math> where <math>M</math> is a set and <math>d</math> is a metric on <math>M</math>, i.e., a function
  
 
:<math>d\,\colon M \times M \to \mathbb{R}</math>
 
:<math>d\,\colon M \times M \to \mathbb{R}</math>
  
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such that for any <math>x, y, z \in M</math>, the following holds:<ref>{{cite book|author=B. Choudhary|title=The Elements of Complex Analysis|url=https://books.google.com/books?id=5K9i2YwgTjYC&pg=PA20|year=1992|publisher=New Age International|isbn=978-81-224-0399-2|page=20}}</ref>
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such that for any <math>x, y, z \in M</math>, the following holds:
  
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<!-- same table is at [[Metric (mathematics)#Definition]] -->
 
 
:{|
 
:{|
 
|-
 
|-
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| 1. || <math>d(x,y) = 0 \iff x = y</math> || [[identity of indiscernibles]]
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| 1. || <math>d(x,y) = 0 \iff x = y</math> || identity of indiscernibles
 
|-
 
|-
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| 2. || <math>d(x,y)  = d(y,x) </math> || [[Symmetric function|symmetry]]
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| 2. || <math>d(x,y)  = d(y,x) </math> || symmetry
 
|-
 
|-
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| 3. || <math>d(x,z) \le d(x,y) + d(y, z)</math>  || [[subadditivity]] or [[triangle inequality]]
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| 3. || <math>d(x,z) \le d(x,y) + d(y, z)</math>  || subadditivity or triangle inequality
 
|}
 
|}
  
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Ignoring mathematical details, for any system of roads and terrains the distance between two locations can be defined as the length of the shortest route connecting those locations. To be a metric there shouldn't be any one-way roads. The triangle inequality expresses the fact that detours aren't shortcuts. If the distance between two points is zero, the two points are indistinguishable from one-another. Many of the examples below can be seen as concrete versions of this general idea.
 
Ignoring mathematical details, for any system of roads and terrains the distance between two locations can be defined as the length of the shortest route connecting those locations. To be a metric there shouldn't be any one-way roads. The triangle inequality expresses the fact that detours aren't shortcuts. If the distance between two points is zero, the two points are indistinguishable from one-another. Many of the examples below can be seen as concrete versions of this general idea.
  
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== Examples of metric spaces == <!-- [[Finite Metric space]] redirects here-->
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== Examples of metric spaces ==  
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* The [[real number]]s with the distance function <math>d(x,y) = | y - x |</math> given by the [[absolute difference]], and, more generally, [[Euclidean space|Euclidean {{mvar|n}}-space]] with the [[Euclidean distance]], are [[complete space|complete]] metric spaces. The [[rational number]]s with the same distance function also form a metric space, but not a complete one.
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* The real numbers with the distance function <math>d(x,y) = | y - x |</math> given by the absolute difference, and, more generally, Euclidean {{mvar|n}}-space with the Euclidean distance, are complete metric spaces. The rational numbers with the same distance function also form a metric space, but not a complete one.
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* The [[positive real numbers]] with distance function <math>d(x,y) =\left| \log(y/x) \right|</math> is a complete metric space.
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* The positive real numbers with distance function <math>d(x,y) =\left| \log(y/x) \right|</math> is a complete metric space.
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* Any [[normed vector space]] is a metric space by defining <math>d(x,y) = \lVert y - x \rVert</math>, see also [[Metric (mathematics)#Metrics on vector spaces|metrics on vector spaces]]. (If such a space is [[Complete metric space|complete]], we call it a [[Banach space]].) Examples:
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* Any normed vector space is a metric space by defining <math>d(x,y) = \lVert y - x \rVert</math>, see also metrics on vector spaces. (If such a space is complete, we call it a Banach space.) Examples:
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** The [[Manhattan norm]] gives rise to the [[Manhattan distance]], where the distance between any two points, or vectors, is the sum of the differences between corresponding coordinates.
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** The Manhattan norm gives rise to the Manhattan distance, where the distance between any two points, or vectors, is the sum of the differences between corresponding coordinates.
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** {{anchor|Mannheim}}The cyclic [[Mannheim metric]] or [[Mannheim distance]] is a modulo variant of the Manhattan metric.<ref name="Huber_1994"/><ref name="Strang-Dammann-Roeckl-Plass_2009"/>
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** The cyclic Mannheim metric or Mannheim distance is a modulo variant of the Manhattan metric.
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** The [[maximum norm]] gives rise to the [[Chebyshev distance]] or chessboard distance, the minimal number of moves a [[king (chess)|chess king]] would take to travel from <math>x</math> to <math>y</math>.
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** The maximum norm gives rise to the Chebyshev distance or chessboard distance, the minimal number of moves a chess king would take to travel from <math>x</math> to <math>y</math>.
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* {{anchor|SNCF}}The [[British Rail]] metric (also called the “post office metric” or the “[[SNCF]] metric”) on a [[normed vector space]] is given by <math>d(x,y) = \lVert x \rVert + \lVert y \rVert</math> for distinct points <math>x</math> and <math>y</math>, and <math>d(x,x) = 0</math>. More generally <math>\lVert \cdot \rVert</math> can be replaced with a function <math>f</math> taking an arbitrary set <math>S</math> to non-negative reals and taking the value <math>0</math> at most once:  then the metric is defined on <math>S</math> by <math>d(x,y) = f(x) + f(y)</math> for distinct points <math>x</math> and <math>y</math>, and {{nowrap|<math>d(x,x) = 0</math>.}} The name alludes to the tendency of railway journeys to proceed via London (or Paris) irrespective of their final destination.
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* The British Rail metric (also called the “post office metric” or the “SNCF metric”) on a normed vector space is given by <math>d(x,y) = \lVert x \rVert + \lVert y \rVert</math> for distinct points <math>x</math> and <math>y</math>, and <math>d(x,x) = 0</math>. More generally <math>\lVert \cdot \rVert</math> can be replaced with a function <math>f</math> taking an arbitrary set <math>S</math> to non-negative reals and taking the value <math>0</math> at most once:  then the metric is defined on <math>S</math> by <math>d(x,y) = f(x) + f(y)</math> for distinct points <math>x</math> and <math>y</math>, and <math>d(x,x) = 0</math>.  The name alludes to the tendency of railway journeys to proceed via London (or Paris) irrespective of their final destination.
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* If <math>(M,d)</math> is a metric space and <math>X</math> is a [[subset]] of {{nowrap|<math>M</math>,}} then <math>(X,d)</math> becomes a metric space by restricting the domain of <math>d</math> to {{nowrap|<math>X \times X</math>.}}
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* If <math>(M,d)</math> is a metric space and <math>X</math> is a subset of <math>M</math>, then <math>(X,d)</math> becomes a metric space by restricting the domain of <math>d</math> to <math>X \times X</math>.
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* The [[discrete metric]], where <math>d(x,y) = 0</math> if <math>x=y</math> and <math>d(x,y) = 1</math> otherwise, is a simple but important example, and can be applied to all sets. This, in particular, shows that for any set, there is always a metric space associated to it. Using this metric, the singleton of any point is an [[open ball]], therefore every subset is open and the space has the [[discrete topology]].
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* The discrete metric, where <math>d(x,y) = 0</math> if <math>x=y</math> and <math>d(x,y) = 1</math> otherwise, is a simple but important example, and can be applied to all sets. This, in particular, shows that for any set, there is always a metric space associated to it. Using this metric, the singleton of any point is an open ball, therefore every subset is open and the space has the discrete topology.
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* A finite metric space is a metric space having a [[finite set|finite]] number of points.  Not every finite metric space can be [[Isometry|isometrically]] embedded in a [[Euclidean space]].<ref>[[Nati Linial|Nathan Linial]]. [https://arxiv.org/abs/math/0304466 ''Finite Metric Spaces—Combinatorics, Geometry and Algorithms'', Proceedings of the ICM, Beijing 2002, vol. 3, pp573–586] {{webarchive|url=https://web.archive.org/web/20180502190916/https://arxiv.org/abs/math/0304466 |date=2018-05-02 }}</ref><ref>[http://kam.mff.cuni.cz/~matousek/metrop.ps''Open problems  on embeddings  of finite  metric  spaces'', edited by Jirīı Matoušek, 2007] {{webarchive|url=https://web.archive.org/web/20101226232112/http://kam.mff.cuni.cz/~matousek/metrop.ps |date=2010-12-26 }}</ref>
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* A finite metric space is a metric space having a finite number of points.  Not every finite metric space can be isometrically embedded in a Euclidean space.
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* The [[hyperbolic geometry#Models of the hyperbolic plane|hyperbolic plane]] is a metric space. More generally:
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* The hyperbolic plane is a metric space. More generally:
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** If <math>M</math> is any [[connectedness|connected]] [[Riemannian manifold]], then we can turn <math>M</math> into a metric space by defining the distance of two points as the [[infimum]] of the lengths of the paths (continuously differentiable [[curve]]s) connecting them.
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** If <math>M</math> is any connected Riemannian manifold, then we can turn <math>M</math> into a metric space by defining the distance of two points as the infimum of the lengths of the paths (continuously differentiable curves) connecting them.
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* If <math>X</math> is some set and <math>M</math> is a metric space, then, the set of all [[bounded function]]s <math>f \colon X \to M</math> (i.e. those functions whose image is a [[bounded subset]] of <math>M</math>) can be turned into a metric space by defining <math>d(f,g) = \sup_{x \in X} d(f(x),g(x))</math> for any two bounded functions <math>f</math> and <math>g</math> (where <math>\sup</math> is [[supremum]]).<ref>Searcóid, [https://books.google.com/books?id=aP37I4QWFRcC&pg=PA107 p. 107].</ref> This metric is called the [[uniform metric]] or supremum metric, and If <math>M</math> is complete, then this [[function space]] is complete as well. If ''X'' is also a topological space, then the set of all bounded [[continuous function (topology)|continuous]] functions from <math>X</math> to <math>M</math> (endowed with the uniform metric), will also be a complete metric if ''M'' is.
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* If <math>X</math> is some set and <math>M</math> is a metric space, then, the set of all bounded functions <math>f \colon X \to M</math> (i.e. those functions whose image is a bounded subset of <math>M</math>) can be turned into a metric space by defining <math>d(f,g) = \sup_{x \in X} d(f(x),g(x))</math> for any two bounded functions <math>f</math> and <math>g</math> (where <math>\sup</math> is supremum). This metric is called the uniform metric or supremum metric, and If <math>M</math> is complete, then this function space is complete as well. If ''X'' is also a topological space, then the set of all bounded continuous functions from <math>X</math> to <math>M</math> (endowed with the uniform metric), will also be a complete metric if ''M'' is.
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* If <math>G</math> is an [[graph theory|undirected connected graph]], then the set <math>V</math> of vertices of <math>G</math> can be turned into a metric space by defining <math>d(x,y)</math> to be the length of the shortest path connecting the vertices <math>x</math> and {{nowrap|<math>y</math>.}} In [[geometric group theory]] this is applied to the [[Cayley graph]] of a group, yielding the [[word metric]].
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* If <math>G</math> is an undirected connected graph, then the set <math>V</math> of vertices of <math>G</math> can be turned into a metric space by defining <math>d(x,y)</math> to be the length of the shortest path connecting the vertices <math>x</math> and <math>y</math>. In geometric group theory this is applied to the Cayley graph of a group, yielding the word metric.
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* [[Graph edit distance]] is a measure of dissimilarity between two [[Graph (discrete mathematics)|graphs]], defined as the minimal number of [[Graph operations|graph edit operations]] required to transform one graph into another.
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* Graph edit distance is a measure of dissimilarity between two graphs, defined as the minimal number of graph edit operations required to transform one graph into another.
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* The [[Levenshtein distance]] is a measure of the dissimilarity between two [[string (computer science)|strings]] <math>u</math> and <math>v</math>, defined as the minimal number of character deletions, insertions, or substitutions required to transform <math>u</math> into <math>v</math>. This can be thought of as a special case of the shortest path metric in a graph and is one example of an [[edit distance]].
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* The Levenshtein distance is a measure of the dissimilarity between two strings <math>u</math> and <math>v</math>, defined as the minimal number of character deletions, insertions, or substitutions required to transform <math>u</math> into <math>v</math>. This can be thought of as a special case of the shortest path metric in a graph and is one example of an edit distance.
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* Given a metric space <math>(X,d)</math> and an increasing [[concave function]] <math>f \colon [0,\infty) \to [0,\infty)</math> such that <math>f(x) = 0</math> if and only if <math>x=0</math>, then <math>f \circ d</math> is also a metric on <math>X</math>.
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* Given a metric space <math>(X,d)</math> and an increasing concave function <math>f \colon [0,\infty) \to [0,\infty)</math> such that <math>f(x) = 0</math> if and only if <math>x=0</math>, then <math>f \circ d</math> is also a metric on <math>X</math>.
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* Given an [[injective function]] <math>f</math> from any set <math>A</math> to a metric space <math>(X,d)</math>, <math>d(f(x), f(y))</math> defines a metric on <math>A</math>.
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* Given an injective function <math>f</math> from any set <math>A</math> to a metric space <math>(X,d)</math>, <math>d(f(x), f(y))</math> defines a metric on <math>A</math>.
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* Using [[T-theory]], the [[tight span]] of a metric space is also a metric space. The tight span is useful in several types of analysis.
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* Using T-theory, the tight span of a metric space is also a metric space. The tight span is useful in several types of analysis.
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* The set of all <math>m</math> by <math>n</math> [[matrix (mathematics)|matrices]] over some [[field (mathematics)|field]] is a metric space with respect to the [[Rank (linear algebra)|rank]] distance <math>d(X,Y) = \mathrm{rank}(Y - X)</math>.
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* The set of all <math>m</math> by <math>n</math> matrices over some field is a metric space with respect to the rank distance <math>d(X,Y) = \mathrm{rank}(Y - X)</math>.
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* The [[Helly metric]] is used in [[game theory]].
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* The Helly metric  is used in game theory.
  
 
== Open and closed sets, topology and convergence ==
 
== Open and closed sets, topology and convergence ==
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Every metric space is a [[topological space]] in a natural manner, and therefore all definitions and theorems about general topological spaces also apply to all metric spaces.
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Every metric space is a topological space in a natural manner, and therefore all definitions and theorems about general topological spaces also apply to all metric spaces.
  
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About any point <math>x</math> in a metric space <math>M</math> we define the '''[[open ball]] of radius <math>r > 0</math> (where <math>r</math> is a real number) about <math>x</math> ''' as the set
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About any point <math>x</math> in a metric space <math>M</math> we define the '''open ball of radius <math>r > 0</math> (where <math>r</math> is a real number) about <math>x</math> ''' as the set
 
:<math>B(x;r) = \{y \in M : d(x,y) < r\}.</math>
 
:<math>B(x;r) = \{y \in M : d(x,y) < r\}.</math>
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These open balls form the [[base (topology)|base]] for a topology on ''M'', making it a [[topological space]].
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These open balls form the base for a topology on ''M'', making it a topological space.
  
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Explicitly, a subset <math>U</math> of <math>M</math> is called '''[[open set|open]]''' if for every <math>x</math> in <math>U</math> there exists an <math>r > 0</math> such that <math>B(x;r)</math> is contained in <math>U</math>. The [[complement (set theory)|complement]] of an open set is called '''[[closed set|closed]]'''. A [[neighborhood (topology)|neighborhood]] of the point <math>x</math> is any subset of <math>M</math> that contains an open ball about <math>x</math> as a subset.
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Explicitly, a subset <math>U</math> of <math>M</math> is called '''open''' if for every <math>x</math> in <math>U</math> there exists an <math>r > 0</math> such that <math>B(x;r)</math> is contained in <math>U</math>. The complement of an open set is called '''closed'''. A neighborhood of the point <math>x</math> is any subset of <math>M</math> that contains an open ball about <math>x</math> as a subset.
  
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A topological space which can arise in this way from a metric space is called a [[metrizable space]].
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A topological space which can arise in this way from a metric space is called a metrizable space.
  
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A [[sequence]] (<math>x_n</math>) in a metric space <math>M</math> is said to [[limit of a sequence|converge]] to the limit <math>x \in M</math> [[if and only if]] for every <math>\varepsilon>0</math>, there exists a natural number ''N'' such that <math>d(x_n,x) < \varepsilon </math> for all <math>n > N</math>. Equivalently, one can use the general definition of convergence available in all topological spaces.
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A sequence (<math>x_n</math>) in a metric space <math>M</math> is said to converge to the limit <math>x \in M</math> if and only if for every <math>\varepsilon>0</math>, there exists a natural number ''N'' such that <math>d(x_n,x) < \varepsilon </math> for all <math>n > N</math>. Equivalently, one can use the general definition of convergence available in all topological spaces.
  
 
A subset <math>A</math> of the metric space <math>M</math> is closed if and only if every sequence in <math>A</math> that converges to a limit in <math>M</math> has its limit in <math>A</math>.
 
A subset <math>A</math> of the metric space <math>M</math> is closed if and only if every sequence in <math>A</math> that converges to a limit in <math>M</math> has its limit in <math>A</math>.
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===Complete spaces===
 
===Complete spaces===
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{{Main|Complete metric space}}
 
  
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A metric space <math>M</math> is said to be '''complete''' if every [[Cauchy sequence]] converges in <math>M</math>. That is to say: if <math>d(x_n, x_m) \to 0</math> as both <math>n</math> and <math>m</math> independently go to infinity, then there is some <math>y\in M</math> with <math>d(x_n, y) \to 0</math>.
 
  
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Every [[Euclidean space]] is complete, as is every closed subset of a complete space. The rational numbers, using the absolute value metric <math>d(x,y) = \vert x - y \vert</math>, are not complete.
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A metric space <math>M</math> is said to be '''complete''' if every Cauchy sequence converges in <math>M</math>. That is to say: if <math>d(x_n, x_m) \to 0</math> as both <math>n</math> and <math>m</math> independently go to infinity, then there is some <math>y\in M</math> with <math>d(x_n, y) \to 0</math>.
  
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Every metric space has a unique (up to [[isometry]]) [[completion (metric space)|completion]], which is a complete space that contains the given space as a [[dense set|dense]] subset. For example, the real numbers are the completion of the rationals.
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Every Euclidean space is complete, as is every closed subset of a complete space. The rational numbers, using the absolute value metric <math>d(x,y) = \vert x - y \vert</math>, are not complete.
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Every metric space has a unique (up to isometry) completion, which is a complete space that contains the given space as a dense subset. For example, the real numbers are the completion of the rationals.
  
 
If <math>X</math> is a complete subset of the metric space <math>M</math>, then <math>X</math> is closed in <math>M</math>. Indeed, a space is complete if and only if it is closed in any containing metric space.
 
If <math>X</math> is a complete subset of the metric space <math>M</math>, then <math>X</math> is closed in <math>M</math>. Indeed, a space is complete if and only if it is closed in any containing metric space.
  
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Every complete metric space is a [[Baire space]].
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Every complete metric space is a Baire space.
  
 
===Bounded and totally bounded spaces===
 
===Bounded and totally bounded spaces===
 
[[File:Diameter of a Set.svg|thumb|Diameter of a set.]]
 
[[File:Diameter of a Set.svg|thumb|Diameter of a set.]]
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{{See also|bounded set}}
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A metric space <math>M</math> is called '''bounded''' if there exists some number <math>r</math>, such that <math>d(x, y) \leq r</math> for all <math>x, y \in M.</math>  The smallest possible such <math>r</math> is called the '''diameter''' of <math>M.</math> The space <math>M</math> is called '''precompact''' or '''totally bounded''' if for every <math>r>0</math> there exist finitely many open balls of radius <math>r</math> whose union covers <math>M.</math> Since the set of the centres of these balls is finite, it has finite diameter, from which it follows (using the triangle inequality) that every totally bounded space is bounded.  The converse does not hold, since any infinite set can be given the discrete metric (one of the examples above) under which it is bounded and yet not totally bounded.
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A metric space <math>M</math> is called '''{{visible anchor|bounded|Metric bounded space}}''' if there exists some number <math>r</math>, such that <math>d(x, y) \leq r</math> for all <math>x, y \in M.</math>  The smallest possible such <math>r</math> is called the '''{{visible anchor|[[diameter]]|Diameter of a metric space}}''' of <math>M.</math> The space <math>M</math> is called '''precompact''' or '''[[totally bounded]]''' if for every <math>r>0</math> there exist finitely many open balls of radius <math>r</math> whose union covers <math>M.</math> Since the set of the centres of these balls is finite, it has finite diameter, from which it follows (using the triangle inequality) that every totally bounded space is bounded.  The converse does not hold, since any infinite set can be given the discrete metric (one of the examples above) under which it is bounded and yet not totally bounded.
 
  
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Note that in the context of [[Interval (mathematics)|interval]]s in the space of [[real number]]s and occasionally regions in a Euclidean space <math>\R^n</math> a bounded set is referred to as "a finite interval" or "finite region". However boundedness should not in general be confused with "finite", which refers to the number of elements, not to how far the set extends; finiteness implies boundedness, but not conversely. Also note that an unbounded subset of <math>\R^n</math> may have a finite [[Lebesgue measure|volume]].
+
Note that in the context of intervals in the space of real numbers and occasionally regions in a Euclidean space <math>\R^n</math> a bounded set is referred to as "a finite interval" or "finite region". However boundedness should not in general be confused with "finite", which refers to the number of elements, not to how far the set extends; finiteness implies boundedness, but not conversely. Also note that an unbounded subset of <math>\R^n</math> may have a finite volume.
  
 
===Compact spaces===
 
===Compact spaces===
−
A metric space <math>M</math> is compact if every sequence in <math>M</math> has a [[subsequence]] that converges to a point in <math>M</math>. This is known as [[sequentially compact space|sequential compactness]] and, in metric spaces (but not in general topological spaces), is equivalent to the topological notions of [[countably compact space|countable compactness]] and [[compact space|compactness]] defined via [[open cover]]s.
+
A metric space <math>M</math> is compact if every sequence in <math>M</math> has a subsequence that converges to a point in <math>M</math>. This is known as sequential compactness and, in metric spaces (but not in general topological spaces), is equivalent to the topological notions of countable compactness and compactness defined via open covers.
  
−
Examples of compact metric spaces include the closed interval <math>[0,1]</math> with the absolute value metric, all metric spaces with finitely many points, and the [[Cantor set]]. Every closed subset of a compact space is itself compact.
+
Examples of compact metric spaces include the closed interval <math>[0,1]</math> with the absolute value metric, all metric spaces with finitely many points, and the Cantor set. Every closed subset of a compact space is itself compact.
  
−
A metric space is compact if and only if it is complete and totally bounded.  This is known as the [[Heine–Borel theorem]]. Note that compactness depends only on the topology, while boundedness depends on the metric.
+
A metric space is compact if and only if it is complete and totally bounded.  This is known as the Heine–Borel theorem. Note that compactness depends only on the topology, while boundedness depends on the metric.
  
−
[[Lebesgue's number lemma]] states that for every open cover of a compact metric space <math>M</math>, there exists a "Lebesgue number" <math>\delta</math> such that every subset of <math>M</math> of [[diameter]] <math>r<\delta</math> is contained in some member of the cover.
+
Lebesgue's number lemma states that for every open cover of a compact metric space <math>M</math>, there exists a "Lebesgue number" <math>\delta</math> such that every subset of <math>M</math> of diameter <math>r<\delta</math> is contained in some member of the cover.
  
−
Every compact metric space is [[second countable]],<ref>{{cite web|url=http://planetmath.org/encyclopedia/CompactMetricSpacesAreSecondCountable.html|title=PlanetMath: a compact metric space is second countable<!-- Bot generated title -->|website=planetmath.org|access-date=2 May 2018|url-status=live|archive-url=https://web.archive.org/web/20090205003114/http://planetmath.org/encyclopedia/CompactMetricSpacesAreSecondCountable.html|archive-date=5 February 2009}}</ref> and is a continuous image of the [[Cantor set]]. (The latter result is due to [[Pavel Alexandrov]] and [[Pavel Samuilovich Urysohn|Urysohn]].)
+
Every compact metric space is second countable, and is a continuous image of the Cantor set. (The latter result is due to Pavel Alexandrov and Urysohn.)
  
 
===Locally compact and proper spaces===
 
===Locally compact and proper spaces===
−
A metric space is said to be [[locally compact]] if every point has a compact neighborhood. Euclidean spaces are locally compact, but infinite-dimensional Banach spaces are not.
+
A metric space is said to be locally compact if every point has a compact neighborhood. Euclidean spaces are locally compact, but infinite-dimensional Banach spaces are not.
  
−
A space is [[proper space|proper]] if every closed ball <math>\{y\, \colon d(x,y)\leq r\}</math> is compact. Proper spaces are locally compact, but the converse is not true in general.
+
A space is proper if every closed ball <math>\{y\, \colon d(x,y)\leq r\}</math> is compact. Proper spaces are locally compact, but the converse is not true in general.
  
 
===Connectedness===
 
===Connectedness===
−
A metric space <math>M</math> is [[connected space|connected]] if the only subsets that are both open and closed are the empty set and <math>M</math> itself.
+
A metric space <math>M</math> is connected if the only subsets that are both open and closed are the empty set and <math>M</math> itself.
  
−
A metric space <math>M</math> is [[connectedness|path connected]] if for any two points <math>x, y \in M</math> there exists a continuous map <math>f\colon [0,1] \to M</math> with <math>f(0)=x</math> and <math>f(1)=y</math>.
+
A metric space <math>M</math> is path connected if for any two points <math>x, y \in M</math> there exists a continuous map <math>f\colon [0,1] \to M</math> with <math>f(0)=x</math> and <math>f(1)=y</math>.
 
Every path connected space is connected, but the converse is not true in general.
 
Every path connected space is connected, but the converse is not true in general.
  
−
There are also local versions of these definitions: [[locally connected space]]s and [[connectedness|locally path connected spaces]].
+
There are also local versions of these definitions: locally connected spaces and locally path connected spaces.
  
−
[[Simply connected space]]s are those that, in a certain sense, do not have "holes".
+
Simply connected spaces are those that, in a certain sense, do not have "holes".
  
 
===Separable spaces===
 
===Separable spaces===
−
A metric space is [[separable space]] if it has a [[countable]] [[dense set|dense]] subset. Typical examples are the real numbers or any Euclidean space. For metric spaces (but not for general topological spaces) separability is equivalent to [[second-countable space|second-countability]] and also to the [[Lindelöf space|Lindelöf]] property.
+
A metric space is separable space if it has a countable dense subset. Typical examples are the real numbers or any Euclidean space. For metric spaces (but not for general topological spaces) separability is equivalent to second-countability and also to the Lindelöf property.
  
 
===Pointed metric spaces===
 
===Pointed metric spaces===
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===Continuous maps===
 
===Continuous maps===
−
{{Main|Continuous function (topology)}}
 
  
−
The map <math>f\,\colon M_1\to M_2</math> is [[continuous map|continuous]]
+
The map <math>f\,\colon M_1\to M_2</math> is continuous
 
if it has one (and therefore all) of the following equivalent properties:
 
if it has one (and therefore all) of the following equivalent properties:
−
;General topological continuity: for every open set <math>U</math> in <math>M_2</math>, the [[preimage]] <math>f^{-1}[U]</math> is open in <math>M_1</math>
+
;General topological continuity: for every open set <math>U</math> in <math>M_2</math>, the preimage <math>f^{-1}[U]</math> is open in <math>M_1</math>
−
:This is the general definition of [[Continuity (topology)|continuity in topology]].
+
:This is the general definition of continuity in topology.
 
;Sequential continuity: if <math>(x_n)</math> is a sequence in <math>M_1</math> that converges to <math>x</math>, then the sequence <math>(f(x_n))</math> converges to <math>f(x)</math> in <math>M_2</math>.
 
;Sequential continuity: if <math>(x_n)</math> is a sequence in <math>M_1</math> that converges to <math>x</math>, then the sequence <math>(f(x_n))</math> converges to <math>f(x)</math> in <math>M_2</math>.
−
:This is [[sequential continuity]], due to [[Eduard Heine]].
+
:This is sequential continuity, due to Eduard Heine.
 
;ε-δ definition: for every <math>x\in M_1</math> and every <math>\varepsilon>0</math> there exists <math>\delta>0</math> such that for all <math>y</math> in <math>M_1</math> we have
 
;ε-δ definition: for every <math>x\in M_1</math> and every <math>\varepsilon>0</math> there exists <math>\delta>0</math> such that for all <math>y</math> in <math>M_1</math> we have
 
::<math>d_1(x,y)<\delta \implies d_2(f(x),f(y))< \varepsilon.</math>
 
::<math>d_1(x,y)<\delta \implies d_2(f(x),f(y))< \varepsilon.</math>
−
:This uses the [[(ε, δ)-definition of limit]], and is due to [[Augustin Louis Cauchy]].
+
:This uses the (ε, δ)-definition of limit, and is due to Augustin Louis Cauchy.
  
 
Moreover, <math>f</math> is continuous if and only if it is continuous on every compact subset of <math>M_1</math>.
 
Moreover, <math>f</math> is continuous if and only if it is continuous on every compact subset of <math>M_1</math>.
  
−
The [[image (mathematics)|image]] of every compact set under a continuous function is compact, and the image of every connected set under a continuous function is connected.
+
The image of every compact set under a continuous function is compact, and the image of every connected set under a continuous function is connected.
  
 
===Uniformly continuous maps===
 
===Uniformly continuous maps===
−
The map <math>f\,\colon M_1\to M_2</math> is [[uniform continuity|uniformly continuous]] if for every <math>\varepsilon>0</math> there exists <math>\delta>0</math> such that
+
The map <math>f\,\colon M_1\to M_2</math> is uniformly continuous if for every <math>\varepsilon>0</math> there exists <math>\delta>0</math> such that
  
 
:<math>d_1(x,y)<\delta \implies d_2(f(x),f(y))< \varepsilon \quad\mbox{for all}\quad x,y\in M_1.</math>
 
:<math>d_1(x,y)<\delta \implies d_2(f(x),f(y))< \varepsilon \quad\mbox{for all}\quad x,y\in M_1.</math>
  
−
Every uniformly continuous map <math>f\,\colon M_1\to M_2</math> is continuous.  The converse is true if <math>M_1</math> is compact ([[Heine–Cantor theorem]]).
+
Every uniformly continuous map <math>f\,\colon M_1\to M_2</math> is continuous.  The converse is true if <math>M_1</math> is compact (Heine–Cantor theorem).
  
 
Uniformly continuous maps turn Cauchy sequences in <math>M_1</math> into Cauchy sequences in <math>M_2</math>. For continuous maps this is generally wrong; for example, a continuous map
 
Uniformly continuous maps turn Cauchy sequences in <math>M_1</math> into Cauchy sequences in <math>M_2</math>. For continuous maps this is generally wrong; for example, a continuous map
Line 171: Line 168:
  
 
===Lipschitz-continuous maps and contractions===
 
===Lipschitz-continuous maps and contractions===
−
{{Main|Lipschitz continuity}}
 
  
−
Given a real number <math>K>0</math>, the map <math>f\,\colon M_1\to M_2</math> is [[Lipschitz continuity|''K''-Lipschitz continuous]] if
+
Given a real number <math>K>0</math>, the map <math>f\,\colon M_1\to M_2</math> is ''K''-Lipschitz continuous if
  
 
:<math>d_2(f(x),f(y))\leq K d_1(x,y)\quad\mbox{for all}\quad x,y\in M_1.</math>
 
:<math>d_2(f(x),f(y))\leq K d_1(x,y)\quad\mbox{for all}\quad x,y\in M_1.</math>
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Every Lipschitz-continuous map is uniformly continuous, but the converse is not true in general.
 
Every Lipschitz-continuous map is uniformly continuous, but the converse is not true in general.
  
−
If <math>K<1</math>, then <math>f</math> is called a [[contraction mapping|contraction]]. Suppose <math>M_2=M_1</math> and <math>M_1</math> is complete. If <math>f</math> is a contraction, then <math>f</math> admits a unique fixed point ([[Banach fixed-point theorem]]). If <math>M_1</math> is compact, the condition can be weakened a bit: <math>f</math> admits a unique fixed point if
+
If <math>K<1</math>, then <math>f</math> is called a contraction. Suppose <math>M_2=M_1</math> and <math>M_1</math> is complete. If <math>f</math> is a contraction, then <math>f</math> admits a unique fixed point (Banach fixed-point theorem). If <math>M_1</math> is compact, the condition can be weakened a bit: <math>f</math> admits a unique fixed point if
  
 
:<math> d(f(x), f(y)) < d(x, y) \quad \mbox{for all} \quad x \ne y \in M_1</math>.
 
:<math> d(f(x), f(y)) < d(x, y) \quad \mbox{for all} \quad x \ne y \in M_1</math>.
  
 
===Isometries===
 
===Isometries===
−
The map <math>f\,\colon M_1\to M_2</math> is an [[isometry]] if
+
The map <math>f\,\colon M_1\to M_2</math> is an isometry if
 
:<math>d_2(f(x),f(y))=d_1(x,y)\quad\mbox{for all}\quad x,y\in M_1</math>
 
:<math>d_2(f(x),f(y))=d_1(x,y)\quad\mbox{for all}\quad x,y\in M_1</math>
−
Isometries are always [[injective]]; the image of a compact or complete set under an isometry is compact or complete, respectively. However, if the isometry is not [[surjective]], then the image of a closed (or open) set need not be closed (or open).
+
Isometries are always injective; the image of a compact or complete set under an isometry is compact or complete, respectively. However, if the isometry is not surjective, then the image of a closed (or open) set need not be closed (or open).
  
 
===Quasi-isometries===
 
===Quasi-isometries===
−
The map <math>f\,\colon M_1\to M_2</math> is a [[quasi-isometry]] if there exist constants <math>A\geq1</math> and <math>B\geq0</math> such that
+
The map <math>f\,\colon M_1\to M_2</math> is a quasi-isometry if there exist constants <math>A\geq1</math> and <math>B\geq0</math> such that
  
 
:<math>\frac{1}{A} d_2(f(x),f(y))-B\leq d_1(x,y)\leq A d_2(f(x),f(y))+B \quad\text{ for all }\quad x,y\in M_1</math>
 
:<math>\frac{1}{A} d_2(f(x),f(y))-B\leq d_1(x,y)\leq A d_2(f(x),f(y))+B \quad\text{ for all }\quad x,y\in M_1</math>
Line 195: Line 191:
 
and a constant <math>C\geq0</math> such that every point in <math>M_2</math> has a distance at most <math>C</math> from some point in the image <math>f(M_1)</math>.
 
and a constant <math>C\geq0</math> such that every point in <math>M_2</math> has a distance at most <math>C</math> from some point in the image <math>f(M_1)</math>.
  
−
Note that a quasi-isometry is not required to be continuous. Quasi-isometries compare the "large-scale structure" of metric spaces; they find use in [[geometric group theory]] in relation to the [[word metric]].
+
Note that a quasi-isometry is not required to be continuous. Quasi-isometries compare the "large-scale structure" of metric spaces; they find use in geometric group theory in relation to the word metric.
  
 
==Notions of metric space equivalence==
 
==Notions of metric space equivalence==
 
Given two metric spaces <math>(M_1, d_1)</math> and <math>(M_2, d_2)</math>:
 
Given two metric spaces <math>(M_1, d_1)</math> and <math>(M_2, d_2)</math>:
  
−
*They are called '''homeomorphic''' (topologically isomorphic) if there exists a [[homeomorphism]] between them (i.e., a [[bijection]] continuous in both directions).
+
*They are called '''homeomorphic''' (topologically isomorphic) if there exists a homeomorphism between them (i.e., a bijection continuous in both directions).
−
*They are called '''uniformic''' (uniformly isomorphic) if there exists a [[uniform isomorphism]] between them (i.e., a [[bijection]] uniformly continuous in both directions).
+
*They are called '''uniformic''' (uniformly isomorphic) if there exists a uniform isomorphism between them (i.e., a bijection uniformly continuous in both directions).
−
*They are called '''isometric''' if there exists a [[bijective]] [[isometry]] between them. In this case, the two metric spaces are essentially identical.
+
*They are called '''isometric''' if there exists a bijective isometry between them. In this case, the two metric spaces are essentially identical.
−
*They are called '''quasi-isometric''' if there exists a [[quasi-isometry]] between them.
+
*They are called '''quasi-isometric''' if there exists a quasi-isometry between them.
  
 
==Topological properties==
 
==Topological properties==
−
Metric spaces are [[paracompact]]<ref>Rudin, Mary Ellen.  [https://www.jstor.org/stable/2035708 A new proof that metric spaces are paracompact] {{webarchive|url=https://web.archive.org/web/20160412015215/http://www.jstor.org/stable/2035708 |date=2016-04-12 }}.  Proceedings of the American Mathematical Society, Vol. 20, No. 2. (Feb., 1969), p. 603.</ref> [[Hausdorff space]]s<ref>{{planetmath reference|id=5838|title=metric spaces are Hausdorff}}</ref> and hence [[normal space|normal]] (indeed they are perfectly normal). An important consequence is that every metric space admits [[partitions of unity]] and that every continuous real-valued function defined on a closed subset of a metric space can be extended to a continuous map on the whole space ([[Tietze extension theorem]]). It is also true that every real-valued [[Lipschitz maps|Lipschitz-continuous]] map defined on a subset of a metric space can be extended to a Lipschitz-continuous map on the whole space.
+
Metric spaces are paracompact Hausdorff spaces and hence normal (indeed they are perfectly normal). An important consequence is that every metric space admits partitions of unity and that every continuous real-valued function defined on a closed subset of a metric space can be extended to a continuous map on the whole space (Tietze extension theorem). It is also true that every real-valued Lipschitz-continuous map defined on a subset of a metric space can be extended to a Lipschitz-continuous map on the whole space.
  
−
Metric spaces are [[first countable]] since one can use balls with rational radius as a neighborhood base.
+
Metric spaces are first countable since one can use balls with rational radius as a neighborhood base.
  
 
The metric topology on a metric space <math>M</math> is the coarsest topology on <math>M</math> relative to which the metric <math>d</math> is a continuous map from the product of <math>M</math> with itself to the non-negative real numbers.
 
The metric topology on a metric space <math>M</math> is the coarsest topology on <math>M</math> relative to which the metric <math>d</math> is a continuous map from the product of <math>M</math> with itself to the non-negative real numbers.
  
 
==Distance between points and sets; Hausdorff distance and Gromov metric==
 
==Distance between points and sets; Hausdorff distance and Gromov metric==
−
A simple way to construct a function separating a point from a closed set (as required for a [[completely regular]] space) is to consider the [[Distance#Distances between sets and between a point and a set|distance between the point and the set]]. If <math>(M,d)</math> is a metric space, <math>S</math> is a [[subset]] of <math>M</math> and <math>x</math> is a point of <math>M</math>, we define the distance from <math>x</math> to <math>S</math> as
+
A simple way to construct a function separating a point from a closed set (as required for a completely regular space) is to consider the distance between the point and the set. If <math>(M,d)</math> is a metric space, <math>S</math> is a subset of <math>M</math> and <math>x</math> is a point of <math>M</math>, we define the distance from <math>x</math> to <math>S</math> as
−
:<math>d(x,S) = \inf\{d(x,s) : s \in S \} </math> where <math>\inf</math> represents the [[infimum]].
+
:<math>d(x,S) = \inf\{d(x,s) : s \in S \} </math> where <math>\inf</math> represents the infimum.
  
−
Then <math>d(x, S)=0</math> if and only if <math>x</math> belongs to the [[closure (topology)|closure]] of <math>S</math>. Furthermore, we have the following generalization of the triangle inequality:
+
Then <math>d(x, S)=0</math> if and only if <math>x</math> belongs to the closure of <math>S</math>. Furthermore, we have the following generalization of the triangle inequality:
 
:<math>d(x,S) \leq d(x,y) + d(y,S),</math>
 
:<math>d(x,S) \leq d(x,y) + d(y,S),</math>
 
which in particular shows that the map <math>x\mapsto d(x,S)</math> is continuous.
 
which in particular shows that the map <math>x\mapsto d(x,S)</math> is continuous.
  
−
Given two subsets <math>S</math> and <math>T</math> of <math>M</math>, we define their [[Hausdorff distance]] to be
+
Given two subsets <math>S</math> and <math>T</math> of <math>M</math>, we define their Hausdorff distance to be
−
:<math>d_H(S,T) = \max \{ \sup\{d(s,T) : s \in S \} , \sup\{ d(t,S) : t \in T \} \} </math> where <math>\sup</math> represents the [[supremum]].
+
:<math>d_H(S,T) = \max \{ \sup\{d(s,T) : s \in S \} , \sup\{ d(t,S) : t \in T \} \} </math> where <math>\sup</math> represents the supremum.
  
 
In general, the Hausdorff distance <math>d_H(S,T)</math> can be infinite. Two sets are close to each other in the Hausdorff distance if every element of either set is close to some element of the other set.
 
In general, the Hausdorff distance <math>d_H(S,T)</math> can be infinite. Two sets are close to each other in the Hausdorff distance if every element of either set is close to some element of the other set.
  
 
The Hausdorff distance <math>d_H</math> turns the set <math>K(M)</math> of all non-empty compact subsets of <math>M</math> into a metric space. One can show that <math>K(M)</math> is complete if <math>M</math> is complete.
 
The Hausdorff distance <math>d_H</math> turns the set <math>K(M)</math> of all non-empty compact subsets of <math>M</math> into a metric space. One can show that <math>K(M)</math> is complete if <math>M</math> is complete.
−
(A different notion of convergence of compact subsets is given by the [[Kuratowski convergence]].)
+
(A different notion of convergence of compact subsets is given by the Kuratowski convergence.)
  
−
One can then define the [[Gromov–Hausdorff convergence|Gromov–Hausdorff distance]] between any two metric spaces by considering the minimal Hausdorff distance of isometrically embedded versions of the two spaces. Using this distance, the class of all (isometry classes of) compact metric spaces becomes a metric space in its own right.
+
One can then define the Gromov–Hausdorff distance between any two metric spaces by considering the minimal Hausdorff distance of isometrically embedded versions of the two spaces. Using this distance, the class of all (isometry classes of) compact metric spaces becomes a metric space in its own right.
  
 
==Product metric spaces==
 
==Product metric spaces==
−
If <math>(M_1,d_1),\ldots,(M_n,d_n)</math> are metric spaces, and <math>N</math> is the [[Euclidean norm]] on <math>\mathbb R^n</math>, then <math>\Bigl(M_1 \times \cdots \times M_n, N(d_1,\ldots,d_n)\Bigr)</math> is a metric space, where the [[product metric]] is defined by
+
If <math>(M_1,d_1),\ldots,(M_n,d_n)</math> are metric spaces, and <math>N</math> is the Euclidean norm on <math>\mathbb R^n</math>, then <math>\Bigl(M_1 \times \cdots \times M_n, N(d_1,\ldots,d_n)\Bigr)</math> is a metric space, where the product metric is defined by
  
 
:<math>N(d_1,\ldots,d_n)\Big((x_1,\ldots,x_n),(y_1,\ldots,y_n)\Big) = N\Big(d_1(x_1,y_1),\ldots,d_n(x_n,y_n)\Big),</math>
 
:<math>N(d_1,\ldots,d_n)\Big((x_1,\ldots,x_n),(y_1,\ldots,y_n)\Big) = N\Big(d_1(x_1,y_1),\ldots,d_n(x_n,y_n)\Big),</math>
  
−
and the induced topology agrees with the [[product topology]].  By the equivalence of norms in finite dimensions, an equivalent metric is obtained if <math>N</math> is the [[Norm (mathematics)#Taxicab norm or Manhattan norm|taxicab norm]], a [[Norm (mathematics)#p-norm|p-norm]], the maximum norm, or any other norm which is non-decreasing as the coordinates of a positive <math>n</math>-tuple increase (yielding the triangle inequality).
+
and the induced topology agrees with the product topology.  By the equivalence of norms in finite dimensions, an equivalent metric is obtained if <math>N</math> is the taxicab norm, a p-norm, the maximum norm, or any other norm which is non-decreasing as the coordinates of a positive <math>n</math>-tuple increase (yielding the triangle inequality).
  
 
Similarly, a countable product of metric spaces can be obtained using the following metric
 
Similarly, a countable product of metric spaces can be obtained using the following metric
Line 241: Line 237:
 
:<math>d(x,y)=\sum_{i=1}^\infty \frac1{2^i}\frac{d_i(x_i,y_i)}{1+d_i(x_i,y_i)}.</math>
 
:<math>d(x,y)=\sum_{i=1}^\infty \frac1{2^i}\frac{d_i(x_i,y_i)}{1+d_i(x_i,y_i)}.</math>
  
−
An uncountable product of metric spaces need not be metrizable. For example, <math>\mathbb{R}^\mathbb{R}</math> is not [[first-countable space|first-countable]] and thus isn't metrizable.
+
An uncountable product of metric spaces need not be metrizable. For example, <math>\mathbb{R}^\mathbb{R}</math> is not first-countable and thus isn't metrizable.
  
 
===Continuity of distance===
 
===Continuity of distance===
−
In the case of a single space <math>(M,d)</math>, the distance map <math>d\colon M\times M \to R^+</math> (from the [[Metric space#Definition|definition]]) is uniformly continuous with respect to any of the above product metrics <math>N(d,d)</math>, and in particular is continuous with respect to the product topology of <math>M\times M</math>.
+
In the case of a single space <math>(M,d)</math>, the distance map <math>d\colon M\times M \to R^+</math> (from the definition) is uniformly continuous with respect to any of the above product metrics <math>N(d,d)</math>, and in particular is continuous with respect to the product topology of <math>M\times M</math>.
  
 
==Quotient metric spaces==
 
==Quotient metric spaces==
−
If ''M'' is a metric space with metric <math>d</math>, and <math>\sim</math> is an [[equivalence relation]] on <math>M</math>, then we can endow the quotient set <math>M/\!\sim</math> with a pseudometric. Given two equivalence classes <math>[x]</math> and <math>[y]</math>, we define
+
If ''M'' is a metric space with metric <math>d</math>, and <math>\sim</math> is an equivalence relation on <math>M</math>, then we can endow the quotient set <math>M/\!\sim</math> with a pseudometric. Given two equivalence classes <math>[x]</math> and <math>[y]</math>, we define
 
:<math>d'([x],[y]) = \inf\{d(p_1,q_1)+d(p_2,q_2)+\dotsb+d(p_{n},q_{n})\}</math>
 
:<math>d'([x],[y]) = \inf\{d(p_1,q_1)+d(p_2,q_2)+\dotsb+d(p_{n},q_{n})\}</math>
  
−
where the [[infimum]] is taken over all finite sequences <math>(p_1, p_2, \dots, p_n)</math> and <math>(q_1, q_2, \dots, q_n)</math> with <math>[p_1]=[x]</math>, <math>[q_n]=[y]</math>, <math>[q_i]=[p_{i+1}], i=1,2,\dots, n-1</math>. In general this will only define a [[pseudometric space|pseudometric]], i.e. <math>d'([x],[y])=0</math> does not necessarily imply that <math>[x]=[y]</math>. However, for some equivalence relations (e.g., those given by gluing together polyhedra along faces), <math>d'</math> is a metric.
+
where the infimum is taken over all finite sequences <math>(p_1, p_2, \dots, p_n)</math> and <math>(q_1, q_2, \dots, q_n)</math> with <math>[p_1]=[x]</math>, <math>[q_n]=[y]</math>, <math>[q_i]=[p_{i+1}], i=1,2,\dots, n-1</math>. In general this will only define a pseudometric, i.e. <math>d'([x],[y])=0</math> does not necessarily imply that <math>[x]=[y]</math>. However, for some equivalence relations (e.g., those given by gluing together polyhedra along faces), <math>d'</math> is a metric.
  
−
The quotient metric <math>d</math> is characterized by the following [[universal property]]. If <math>f\,\colon(M,d)\to(X,\delta)</math> is a [[metric map]] between metric spaces (that is, <math>\delta(f(x),f(y))\le d(x,y)</math> for all <math>x</math>, <math>y</math>) satisfying <math>f(x)=f(y)</math> whenever <math>x\sim y,</math> then the induced function <math>\overline{f}\,\colon M/\!\sim\to X</math>, given by <math>\overline{f}([x])=f(x)</math>, is a metric map <math>\overline{f}\,\colon (M/\!\sim,d')\to (X,\delta).</math>
+
The quotient metric <math>d</math> is characterized by the following universal property. If <math>f\,\colon(M,d)\to(X,\delta)</math> is a metric map between metric spaces (that is, <math>\delta(f(x),f(y))\le d(x,y)</math> for all <math>x</math>, <math>y</math>) satisfying <math>f(x)=f(y)</math> whenever <math>x\sim y,</math> then the induced function <math>\overline{f}\,\colon M/\!\sim\to X</math>, given by <math>\overline{f}([x])=f(x)</math>, is a metric map <math>\overline{f}\,\colon (M/\!\sim,d')\to (X,\delta).</math>
  
−
A topological space is [[sequential space|sequential]] if and only if it is a quotient of a metric space.<ref>Goreham, Anthony. [http://at.yorku.ca/p/a/a/o/51.pdf Sequential convergence in Topological Spaces] {{webarchive|url=https://web.archive.org/web/20110604232111/http://at.yorku.ca/p/a/a/o/51.pdf |date=2011-06-04 }}. Honours' Dissertation, Queen's College, Oxford (April, 2001), p. 14</ref>
+
A topological space is sequential if and only if it is a quotient of a metric space.
  
 
==Generalizations of metric spaces==
 
==Generalizations of metric spaces==
−
* Every metric space is a [[uniform space]] in a natural manner, and every uniform space is naturally a [[topological space]]. Uniform and topological spaces can therefore be regarded as generalizations of metric spaces.
+
*Every metric space is a uniform space in a natural manner, and every uniform space is naturally a topological space. Uniform and topological spaces can therefore be regarded as generalizations of metric spaces.
−
* Relaxing the requirement that the distance between two distinct points be non-zero leads to the concepts of a [[pseudometric space]] or a dislocated metric space.<ref name="HitzlerSeda2016">{{cite book|author1=Pascal Hitzler|author2=Anthony Seda|title=Mathematical Aspects of Logic Programming Semantics|url=https://books.google.com/books?id=VKPMBQAAQBAJ|date=19 April 2016|publisher=CRC Press|isbn=978-1-4398-2962-2|author1-link=Pascal Hitzler}}</ref> Removing the requirement of symmetry, we arrive at a [[quasimetric space]]. Replacing the triangle inequality with a weaker form leads to [[semimetric space]]s.
+
* Relaxing the requirement that the distance between two distinct points be non-zero leads to the concepts of a pseudometric space or a dislocated metric space.[12] Removing the requirement of symmetry, we arrive at a quasimetric space. Replacing the triangle inequality with a weaker form leads to semimetric spaces.
−
* If the distance function takes values in the [[extended real number line]] <math>\mathbb R\cup\{+\infty\}</math>, but otherwise satisfies the conditions of a metric, then it is called an ''extended metric'' and the corresponding space is called an ''<math>\infty</math>-metric space''. If the distance function takes values in some (suitable) ordered set (and the triangle inequality is adjusted accordingly), then we arrive at the notion of ''generalized ultrametric''.<ref name="HitzlerSeda2016"/>
+
* If the distance function takes values in the extended real number line <math>\mathbb R\cup\{+\infty\}</math>, but otherwise satisfies the conditions of a metric, then it is called an ''extended metric'' and the corresponding space is called an ''<math>\infty</math>-metric space''. If the distance function takes values in some (suitable) ordered set (and the triangle inequality is adjusted accordingly), then we arrive at the notion of ''generalized ultrametric''.
−
* [[Approach space]]s are a generalization of metric spaces, based on point-to-set distances, instead of point-to-point distances.
+
* Approach spaces are a generalization of metric spaces, based on point-to-set distances, instead of point-to-point distances.
−
* A [[continuity space]] is a generalization of metric spaces and [[poset]]s, that can be used to unify the notions of metric spaces and [[Domain theory|domain]]s.
+
* A continuity space is a generalization of metric spaces and posets, that can be used to unify the notions of metric spaces and domains.
−
* A partial metric space is intended to be the least generalisation of the notion of a metric space, such that the distance of each point from itself is no longer necessarily zero.<ref>{{cite web|url=http://www.dcs.warwick.ac.uk/pmetric/|title=Partial metrics : welcome|website=www.dcs.warwick.ac.uk|access-date=2 May 2018|url-status=live|archive-url=https://web.archive.org/web/20170727003912/http://www.dcs.warwick.ac.uk/pmetric/|archive-date=27 July 2017}}</ref>
+
* A partial metric space is intended to be the least generalisation of the notion of a metric space, such that the distance of each point from itself is no longer necessarily zero.
  
 
=== Metric spaces as enriched categories ===
 
=== Metric spaces as enriched categories ===
−
The ordered set <math>(\mathbb{R},\geq)</math> can be seen as a [[Category (mathematics)|category]] by requesting exactly one [[morphism]] <math>a\to b</math> if <math>a\geq b</math> and none otherwise. By using <math>+</math> as the [[tensor product]] and <math>0</math> as the [[Identity element|identity]], it becomes a [[monoidal category]] <math>R^*</math>.
+
The ordered set <math>(\mathbb{R},\geq)</math> can be seen as a category by requesting exactly one morphism <math>a\to b</math> if <math>a\geq b</math> and none otherwise. By using <math>+</math> as the tensor product and <math>0</math> as the identity, it becomes a monoidal category <math>R^*</math>.
−
Every metric space <math>(M,d)</math> can now be viewed as a category <math>M^*</math> [[enriched category|enriched]] over <math>R^*</math>:
+
Every metric space <math>(M,d)</math> can now be viewed as a category <math>M^*</math> enriched over <math>R^*</math>:
 
* Set <math>\operatorname{Ob}(M^*):=M</math>
 
* Set <math>\operatorname{Ob}(M^*):=M</math>
 
* For each <math>X,Y\in M</math> set <math>\operatorname{Hom}(X,Y):=d(X,Y)\in \operatorname{Ob}(R^*)</math>
 
* For each <math>X,Y\in M</math> set <math>\operatorname{Hom}(X,Y):=d(X,Y)\in \operatorname{Ob}(R^*)</math>
 
* The composition morphism <math>\operatorname{Hom}(Y,Z)\otimes \operatorname{Hom}(X,Y)\to \operatorname{Hom}(X,Z)</math> will be the unique morphism in <math>R^*</math> given from the triangle inequality <math>d(y,z)+d(x,y)\geq d(x,z)</math>
 
* The composition morphism <math>\operatorname{Hom}(Y,Z)\otimes \operatorname{Hom}(X,Y)\to \operatorname{Hom}(X,Z)</math> will be the unique morphism in <math>R^*</math> given from the triangle inequality <math>d(y,z)+d(x,y)\geq d(x,z)</math>
 
* The identity morphism <math>0\to \operatorname{Hom}(X,X)</math> will be the unique morphism given from the fact that <math>0\geq d(X,X)</math>.
 
* The identity morphism <math>0\to \operatorname{Hom}(X,X)</math> will be the unique morphism given from the fact that <math>0\geq d(X,X)</math>.
−
* Since <math>R^*</math> is a poset, all [[Diagram (category theory)|diagrams]] that are required for an enriched category commute automatically.
+
* Since <math>R^*</math> is a poset, all diagrams that are required for an enriched category commute automatically.
  
 
==Licensing==
 
==Licensing==
 
Content obtained and/or adapted from:
 
Content obtained and/or adapted from:
 
* [https://en.wikipedia.org/wiki/Metric_space Metric space, Wikipedia] under a CC BY-SA license
 
* [https://en.wikipedia.org/wiki/Metric_space Metric space, Wikipedia] under a CC BY-SA license

Latest revision as of 22:13, 26 October 2021

In mathematics, a metric space is a set together with a metric on the set. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally:

  • the distance from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} is zero if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} are the same point,
  • the distance between two distinct points is positive,
  • the distance from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} is the same as the distance from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , and
  • the distance from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} is less than or equal to the distance from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} via any third point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} .

A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces.

The most familiar metric space is 3-dimensional Euclidean space. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities. Some of non-geometric metric spaces include spaces of finite strings (finite sequences of symbols from a predefined alphabet) equipped with e.g. a Hamming's or Levenshtein distance, a space of subsets of any metric space equipped with Hausdorff distance, a space of real functions integrable on a unit interval with an integral metric Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(f,g)=\int_{x=0}^{x=1}\left\vert f(x)-g(x)\right\vert\,dx} or probabilistic spaces on any chosen metric space equipped with Wasserstein metric. See also the section Examples of metric spaces.

Definition

A metric space is an ordered pair Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M,d)} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is a set and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} is a metric on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , i.e., a function

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\,\colon M \times M \to \mathbb{R}}

such that for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, y, z \in M} , the following holds:

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y) = 0 \iff x = y} identity of indiscernibles
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y) = d(y,x) } symmetry
3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,z) \le d(x,y) + d(y, z)} subadditivity or triangle inequality

Given the above three axioms, we also have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y) \ge 0 } for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, y \in M} . This is deduced as follows (from the top to the bottom):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y) + d(y,x) \ge d(x,x)} by triangle inequality
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y) + d(x,y) \ge d(x,x)} by symmetry
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2d(x,y) \ge 0} by identity of indiscernibles
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y) \ge 0} we have non-negativity

The function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} is also called distance function or simply distance. Often, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} is omitted and one just writes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} for a metric space if it is clear from the context what metric is used.

Ignoring mathematical details, for any system of roads and terrains the distance between two locations can be defined as the length of the shortest route connecting those locations. To be a metric there shouldn't be any one-way roads. The triangle inequality expresses the fact that detours aren't shortcuts. If the distance between two points is zero, the two points are indistinguishable from one-another. Many of the examples below can be seen as concrete versions of this general idea.

Examples of metric spaces

  • The real numbers with the distance function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y) = | y - x |} given by the absolute difference, and, more generally, Euclidean n-space with the Euclidean distance, are complete metric spaces. The rational numbers with the same distance function also form a metric space, but not a complete one.
  • The positive real numbers with distance function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y) =\left| \log(y/x) \right|} is a complete metric space.
  • Any normed vector space is a metric space by defining , see also metrics on vector spaces. (If such a space is complete, we call it a Banach space.) Examples:
    • The Manhattan norm gives rise to the Manhattan distance, where the distance between any two points, or vectors, is the sum of the differences between corresponding coordinates.
    • The cyclic Mannheim metric or Mannheim distance is a modulo variant of the Manhattan metric.
    • The maximum norm gives rise to the Chebyshev distance or chessboard distance, the minimal number of moves a chess king would take to travel from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} .
  • The British Rail metric (also called the “post office metric” or the “SNCF metric”) on a normed vector space is given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y) = \lVert x \rVert + \lVert y \rVert} for distinct points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,x) = 0} . More generally Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lVert \cdot \rVert} can be replaced with a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} taking an arbitrary set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} to non-negative reals and taking the value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} at most once: then the metric is defined on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y) = f(x) + f(y)} for distinct points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,x) = 0} . The name alludes to the tendency of railway journeys to proceed via London (or Paris) irrespective of their final destination.
  • If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M,d)} is a metric space and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a subset of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,d)} becomes a metric space by restricting the domain of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \times X} .
  • The discrete metric, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y) = 0} if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=y} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y) = 1} otherwise, is a simple but important example, and can be applied to all sets. This, in particular, shows that for any set, there is always a metric space associated to it. Using this metric, the singleton of any point is an open ball, therefore every subset is open and the space has the discrete topology.
  • A finite metric space is a metric space having a finite number of points. Not every finite metric space can be isometrically embedded in a Euclidean space.
  • The hyperbolic plane is a metric space. More generally:
    • If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is any connected Riemannian manifold, then we can turn Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} into a metric space by defining the distance of two points as the infimum of the lengths of the paths (continuously differentiable curves) connecting them.
  • If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is some set and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is a metric space, then, the set of all bounded functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \colon X \to M} (i.e. those functions whose image is a bounded subset of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} ) can be turned into a metric space by defining Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(f,g) = \sup_{x \in X} d(f(x),g(x))} for any two bounded functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} (where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sup} is supremum). This metric is called the uniform metric or supremum metric, and If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is complete, then this function space is complete as well. If X is also a topological space, then the set of all bounded continuous functions from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} (endowed with the uniform metric), will also be a complete metric if M is.
  • If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is an undirected connected graph, then the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} of vertices of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} can be turned into a metric space by defining Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y)} to be the length of the shortest path connecting the vertices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} . In geometric group theory this is applied to the Cayley graph of a group, yielding the word metric.
  • Graph edit distance is a measure of dissimilarity between two graphs, defined as the minimal number of graph edit operations required to transform one graph into another.
  • The Levenshtein distance is a measure of the dissimilarity between two strings Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} , defined as the minimal number of character deletions, insertions, or substitutions required to transform Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} into . This can be thought of as a special case of the shortest path metric in a graph and is one example of an edit distance.
  • Given a metric space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,d)} and an increasing concave function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \colon [0,\infty) \to [0,\infty)} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = 0} if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=0} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \circ d} is also a metric on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} .
  • Given an injective function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} from any set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} to a metric space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,d)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(f(x), f(y))} defines a metric on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} .
  • Using T-theory, the tight span of a metric space is also a metric space. The tight span is useful in several types of analysis.
  • The set of all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} matrices over some field is a metric space with respect to the rank distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(X,Y) = \mathrm{rank}(Y - X)} .
  • The Helly metric is used in game theory.

Open and closed sets, topology and convergence

Every metric space is a topological space in a natural manner, and therefore all definitions and theorems about general topological spaces also apply to all metric spaces.

About any point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} in a metric space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} we define the open ball of radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r > 0} (where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is a real number) about Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} as the set

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(x;r) = \{y \in M : d(x,y) < r\}.}

These open balls form the base for a topology on M, making it a topological space.

Explicitly, a subset Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is called open if for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} there exists an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r > 0} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(x;r)} is contained in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} . The complement of an open set is called closed. A neighborhood of the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is any subset of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} that contains an open ball about Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} as a subset.

A topological space which can arise in this way from a metric space is called a metrizable space.

A sequence (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_n} ) in a metric space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is said to converge to the limit Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in M} if and only if for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon>0} , there exists a natural number N such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x_n,x) < \varepsilon } for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n > N} . Equivalently, one can use the general definition of convergence available in all topological spaces.

A subset Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} of the metric space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is closed if and only if every sequence in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} that converges to a limit in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} has its limit in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} .

Types of metric spaces

Complete spaces

A metric space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is said to be complete if every Cauchy sequence converges in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} . That is to say: if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x_n, x_m) \to 0} as both Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} independently go to infinity, then there is some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\in M} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x_n, y) \to 0} .

Every Euclidean space is complete, as is every closed subset of a complete space. The rational numbers, using the absolute value metric Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y) = \vert x - y \vert} , are not complete.

Every metric space has a unique (up to isometry) completion, which is a complete space that contains the given space as a dense subset. For example, the real numbers are the completion of the rationals.

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a complete subset of the metric space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is closed in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} . Indeed, a space is complete if and only if it is closed in any containing metric space.

Every complete metric space is a Baire space.

Bounded and totally bounded spaces

Diameter of a set.

A metric space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is called bounded if there exists some number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} , such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x, y) \leq r} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, y \in M.} The smallest possible such Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is called the diameter of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M.} The space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is called precompact or totally bounded if for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r>0} there exist finitely many open balls of radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} whose union covers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M.} Since the set of the centres of these balls is finite, it has finite diameter, from which it follows (using the triangle inequality) that every totally bounded space is bounded. The converse does not hold, since any infinite set can be given the discrete metric (one of the examples above) under which it is bounded and yet not totally bounded.

Note that in the context of intervals in the space of real numbers and occasionally regions in a Euclidean space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^n} a bounded set is referred to as "a finite interval" or "finite region". However boundedness should not in general be confused with "finite", which refers to the number of elements, not to how far the set extends; finiteness implies boundedness, but not conversely. Also note that an unbounded subset of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^n} may have a finite volume.

Compact spaces

A metric space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is compact if every sequence in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} has a subsequence that converges to a point in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} . This is known as sequential compactness and, in metric spaces (but not in general topological spaces), is equivalent to the topological notions of countable compactness and compactness defined via open covers.

Examples of compact metric spaces include the closed interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0,1]} with the absolute value metric, all metric spaces with finitely many points, and the Cantor set. Every closed subset of a compact space is itself compact.

A metric space is compact if and only if it is complete and totally bounded. This is known as the Heine–Borel theorem. Note that compactness depends only on the topology, while boundedness depends on the metric.

Lebesgue's number lemma states that for every open cover of a compact metric space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , there exists a "Lebesgue number" Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} such that every subset of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} of diameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r<\delta} is contained in some member of the cover.

Every compact metric space is second countable, and is a continuous image of the Cantor set. (The latter result is due to Pavel Alexandrov and Urysohn.)

Locally compact and proper spaces

A metric space is said to be locally compact if every point has a compact neighborhood. Euclidean spaces are locally compact, but infinite-dimensional Banach spaces are not.

A space is proper if every closed ball Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{y\, \colon d(x,y)\leq r\}} is compact. Proper spaces are locally compact, but the converse is not true in general.

Connectedness

A metric space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is connected if the only subsets that are both open and closed are the empty set and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} itself.

A metric space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is path connected if for any two points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, y \in M} there exists a continuous map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\colon [0,1] \to M} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(0)=x} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(1)=y} . Every path connected space is connected, but the converse is not true in general.

There are also local versions of these definitions: locally connected spaces and locally path connected spaces.

Simply connected spaces are those that, in a certain sense, do not have "holes".

Separable spaces

A metric space is separable space if it has a countable dense subset. Typical examples are the real numbers or any Euclidean space. For metric spaces (but not for general topological spaces) separability is equivalent to second-countability and also to the Lindelöf property.

Pointed metric spaces

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a metric space and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0\in X} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,x_0)} is called a pointed metric space, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} is called a distinguished point. Note that a pointed metric space is just a nonempty metric space with attention drawn to its distinguished point, and that any nonempty metric space can be viewed as a pointed metric space. The distinguished point is sometimes denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} due to its similar behavior to zero in certain contexts.

Types of maps between metric spaces

Suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M_1,d_1)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M_2,d_2)} are two metric spaces.

Continuous maps

The map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\,\colon M_1\to M_2} is continuous if it has one (and therefore all) of the following equivalent properties:

General topological continuity
for every open set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_2} , the preimage Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}[U]} is open in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_1}
This is the general definition of continuity in topology.
Sequential continuity
if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_n)} is a sequence in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_1} that converges to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , then the sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f(x_n))} converges to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_2} .
This is sequential continuity, due to Eduard Heine.
ε-δ definition
for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in M_1} and every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon>0} there exists Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta>0} such that for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_1} we have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_1(x,y)<\delta \implies d_2(f(x),f(y))< \varepsilon.}
This uses the (ε, δ)-definition of limit, and is due to Augustin Louis Cauchy.

Moreover, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is continuous if and only if it is continuous on every compact subset of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_1} .

The image of every compact set under a continuous function is compact, and the image of every connected set under a continuous function is connected.

Uniformly continuous maps

The map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\,\colon M_1\to M_2} is uniformly continuous if for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon>0} there exists Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta>0} such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_1(x,y)<\delta \implies d_2(f(x),f(y))< \varepsilon \quad\mbox{for all}\quad x,y\in M_1.}

Every uniformly continuous map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\,\colon M_1\to M_2} is continuous. The converse is true if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_1} is compact (Heine–Cantor theorem).

Uniformly continuous maps turn Cauchy sequences in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_1} into Cauchy sequences in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_2} . For continuous maps this is generally wrong; for example, a continuous map from the open interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,1)} onto the real line turns some Cauchy sequences into unbounded sequences.

Lipschitz-continuous maps and contractions

Given a real number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K>0} , the map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\,\colon M_1\to M_2} is K-Lipschitz continuous if

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_2(f(x),f(y))\leq K d_1(x,y)\quad\mbox{for all}\quad x,y\in M_1.}

Every Lipschitz-continuous map is uniformly continuous, but the converse is not true in general.

If , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is called a contraction. Suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_2=M_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_1} is complete. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is a contraction, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} admits a unique fixed point (Banach fixed-point theorem). If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_1} is compact, the condition can be weakened a bit: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} admits a unique fixed point if

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(f(x), f(y)) < d(x, y) \quad \mbox{for all} \quad x \ne y \in M_1} .

Isometries

The map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\,\colon M_1\to M_2} is an isometry if

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_2(f(x),f(y))=d_1(x,y)\quad\mbox{for all}\quad x,y\in M_1}

Isometries are always injective; the image of a compact or complete set under an isometry is compact or complete, respectively. However, if the isometry is not surjective, then the image of a closed (or open) set need not be closed (or open).

Quasi-isometries

The map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\,\colon M_1\to M_2} is a quasi-isometry if there exist constants Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\geq1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B\geq0} such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{A} d_2(f(x),f(y))-B\leq d_1(x,y)\leq A d_2(f(x),f(y))+B \quad\text{ for all }\quad x,y\in M_1}

and a constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\geq0} such that every point in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_2} has a distance at most Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} from some point in the image Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(M_1)} .

Note that a quasi-isometry is not required to be continuous. Quasi-isometries compare the "large-scale structure" of metric spaces; they find use in geometric group theory in relation to the word metric.

Notions of metric space equivalence

Given two metric spaces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M_1, d_1)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M_2, d_2)} :

  • They are called homeomorphic (topologically isomorphic) if there exists a homeomorphism between them (i.e., a bijection continuous in both directions).
  • They are called uniformic (uniformly isomorphic) if there exists a uniform isomorphism between them (i.e., a bijection uniformly continuous in both directions).
  • They are called isometric if there exists a bijective isometry between them. In this case, the two metric spaces are essentially identical.
  • They are called quasi-isometric if there exists a quasi-isometry between them.

Topological properties

Metric spaces are paracompact Hausdorff spaces and hence normal (indeed they are perfectly normal). An important consequence is that every metric space admits partitions of unity and that every continuous real-valued function defined on a closed subset of a metric space can be extended to a continuous map on the whole space (Tietze extension theorem). It is also true that every real-valued Lipschitz-continuous map defined on a subset of a metric space can be extended to a Lipschitz-continuous map on the whole space.

Metric spaces are first countable since one can use balls with rational radius as a neighborhood base.

The metric topology on a metric space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is the coarsest topology on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} relative to which the metric Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} is a continuous map from the product of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} with itself to the non-negative real numbers.

Distance between points and sets; Hausdorff distance and Gromov metric

A simple way to construct a function separating a point from a closed set (as required for a completely regular space) is to consider the distance between the point and the set. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M,d)} is a metric space, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is a subset of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is a point of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , we define the distance from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,S) = \inf\{d(x,s) : s \in S \} } where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \inf} represents the infimum.

Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x, S)=0} if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} belongs to the closure of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} . Furthermore, we have the following generalization of the triangle inequality:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,S) \leq d(x,y) + d(y,S),}

which in particular shows that the map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\mapsto d(x,S)} is continuous.

Given two subsets Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , we define their Hausdorff distance to be

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_H(S,T) = \max \{ \sup\{d(s,T) : s \in S \} , \sup\{ d(t,S) : t \in T \} \} } where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sup} represents the supremum.

In general, the Hausdorff distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_H(S,T)} can be infinite. Two sets are close to each other in the Hausdorff distance if every element of either set is close to some element of the other set.

The Hausdorff distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_H} turns the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K(M)} of all non-empty compact subsets of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} into a metric space. One can show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K(M)} is complete if is complete. (A different notion of convergence of compact subsets is given by the Kuratowski convergence.)

One can then define the Gromov–Hausdorff distance between any two metric spaces by considering the minimal Hausdorff distance of isometrically embedded versions of the two spaces. Using this distance, the class of all (isometry classes of) compact metric spaces becomes a metric space in its own right.

Product metric spaces

If are metric spaces, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is the Euclidean norm on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb R^n} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Bigl(M_1 \times \cdots \times M_n, N(d_1,\ldots,d_n)\Bigr)} is a metric space, where the product metric is defined by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N(d_1,\ldots,d_n)\Big((x_1,\ldots,x_n),(y_1,\ldots,y_n)\Big) = N\Big(d_1(x_1,y_1),\ldots,d_n(x_n,y_n)\Big),}

and the induced topology agrees with the product topology. By the equivalence of norms in finite dimensions, an equivalent metric is obtained if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is the taxicab norm, a p-norm, the maximum norm, or any other norm which is non-decreasing as the coordinates of a positive Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -tuple increase (yielding the triangle inequality).

Similarly, a countable product of metric spaces can be obtained using the following metric

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y)=\sum_{i=1}^\infty \frac1{2^i}\frac{d_i(x_i,y_i)}{1+d_i(x_i,y_i)}.}

An uncountable product of metric spaces need not be metrizable. For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^\mathbb{R}} is not first-countable and thus isn't metrizable.

Continuity of distance

In the case of a single space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M,d)} , the distance map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\colon M\times M \to R^+} (from the definition) is uniformly continuous with respect to any of the above product metrics Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N(d,d)} , and in particular is continuous with respect to the product topology of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\times M} .

Quotient metric spaces

If M is a metric space with metric Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim} is an equivalence relation on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , then we can endow the quotient set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M/\!\sim} with a pseudometric. Given two equivalence classes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x]} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [y]} , we define

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d'([x],[y]) = \inf\{d(p_1,q_1)+d(p_2,q_2)+\dotsb+d(p_{n},q_{n})\}}

where the infimum is taken over all finite sequences Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (p_1, p_2, \dots, p_n)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (q_1, q_2, \dots, q_n)} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [p_1]=[x]} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [q_n]=[y]} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [q_i]=[p_{i+1}], i=1,2,\dots, n-1} . In general this will only define a pseudometric, i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d'([x],[y])=0} does not necessarily imply that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x]=[y]} . However, for some equivalence relations (e.g., those given by gluing together polyhedra along faces), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d'} is a metric.

The quotient metric Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} is characterized by the following universal property. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\,\colon(M,d)\to(X,\delta)} is a metric map between metric spaces (that is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta(f(x),f(y))\le d(x,y)} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} ) satisfying Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=f(y)} whenever Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\sim y,} then the induced function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{f}\,\colon M/\!\sim\to X} , given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{f}([x])=f(x)} , is a metric map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{f}\,\colon (M/\!\sim,d')\to (X,\delta).}

A topological space is sequential if and only if it is a quotient of a metric space.

Generalizations of metric spaces

  • Every metric space is a uniform space in a natural manner, and every uniform space is naturally a topological space. Uniform and topological spaces can therefore be regarded as generalizations of metric spaces.
  • Relaxing the requirement that the distance between two distinct points be non-zero leads to the concepts of a pseudometric space or a dislocated metric space.[12] Removing the requirement of symmetry, we arrive at a quasimetric space. Replacing the triangle inequality with a weaker form leads to semimetric spaces.
  • If the distance function takes values in the extended real number line Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb R\cup\{+\infty\}} , but otherwise satisfies the conditions of a metric, then it is called an extended metric and the corresponding space is called an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty} -metric space. If the distance function takes values in some (suitable) ordered set (and the triangle inequality is adjusted accordingly), then we arrive at the notion of generalized ultrametric.
  • Approach spaces are a generalization of metric spaces, based on point-to-set distances, instead of point-to-point distances.
  • A continuity space is a generalization of metric spaces and posets, that can be used to unify the notions of metric spaces and domains.
  • A partial metric space is intended to be the least generalisation of the notion of a metric space, such that the distance of each point from itself is no longer necessarily zero.

Metric spaces as enriched categories

The ordered set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{R},\geq)} can be seen as a category by requesting exactly one morphism Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\to b} if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\geq b} and none otherwise. By using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +} as the tensor product and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} as the identity, it becomes a monoidal category Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^*} . Every metric space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M,d)} can now be viewed as a category Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M^*} enriched over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^*} :

  • Set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Ob}(M^*):=M}
  • For each set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Hom}(X,Y):=d(X,Y)\in \operatorname{Ob}(R^*)}
  • The composition morphism Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Hom}(Y,Z)\otimes \operatorname{Hom}(X,Y)\to \operatorname{Hom}(X,Z)} will be the unique morphism in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^*} given from the triangle inequality Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(y,z)+d(x,y)\geq d(x,z)}
  • The identity morphism Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\to \operatorname{Hom}(X,X)} will be the unique morphism given from the fact that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\geq d(X,X)} .
  • Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^*} is a poset, all diagrams that are required for an enriched category commute automatically.

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