Difference between revisions of "Subspaces of Metric Spaces"

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(Created page with "{{Redirect|Induced topology|the topology generated by a family of functions|Initial topology}} In topology and related areas of mathematics, a '''subspace''' of a t...")
 
 
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{{Redirect|Induced topology|the topology generated by a family of functions|Initial topology}}
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= Subspace topology=
 
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In topology and related areas of mathematics, a '''subspace''' of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the '''subspace topology''' (or the '''relative topology''', or the '''induced topology''', or the '''trace topology''').
In [[topology]] and related areas of [[mathematics]], a '''subspace''' of a [[topological space]] ''X'' is a [[subset]] ''S'' of ''X'' which is equipped with a [[Topological_space#Definitions|topology]] induced from that of ''X'' called the '''subspace topology''' (or the '''relative topology''', or the '''induced topology''', or the '''trace topology''').
 
  
 
== Definition ==
 
== Definition ==
  
Given a topological space <math>(X, \tau)</math> and a [[subset]] <math>S</math> of <math>X</math>, the '''subspace topology''' on <math>S</math> is defined by
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Given a topological space <math>(X, \tau)</math> and a subset <math>S</math> of <math>X</math>, the '''subspace topology''' on <math>S</math> is defined by
 
:<math>\tau_S = \lbrace S \cap U \mid U \in \tau \rbrace.</math>
 
:<math>\tau_S = \lbrace S \cap U \mid U \in \tau \rbrace.</math>
That is, a subset of <math>S</math> is open in the subspace topology [[if and only if]] it is the [[intersection (set theory)|intersection]] of <math>S</math> with an [[open set]] in <math>(X, \tau)</math>. If <math>S</math> is equipped with the subspace topology then it is a topological space in its own right, and is called a '''subspace''' of <math>(X, \tau)</math>. Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated.
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That is, a subset of <math>S</math> is open in the subspace topology if and only if it is the intersection of <math>S</math> with an open set in <math>(X, \tau)</math>. If <math>S</math> is equipped with the subspace topology then it is a topological space in its own right, and is called a '''subspace''' of <math>(X, \tau)</math>. Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated.
  
Alternatively we can define the subspace topology for a subset <math>S</math> of <math>X</math> as the [[coarsest topology]] for which the [[inclusion map]]
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Alternatively we can define the subspace topology for a subset <math>S</math> of <math>X</math> as the coarsest topology for which the inclusion map
 
:<math>\iota: S \hookrightarrow X</math>
 
:<math>\iota: S \hookrightarrow X</math>
is [[continuous (topology)|continuous]].
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is continuous.
  
More generally, suppose  <math>\iota</math> is an [[Injective function|injection]] from a set <math>S</math> to a topological space <math>X</math>. Then the subspace topology on <math>S</math> is defined as the coarsest topology for which <math>\iota</math> is continuous. The open sets in this topology are precisely the ones of the form <math>\iota^{-1}(U)</math> for <math>U</math> open in <math>X</math>. <math>S</math> is then [[homeomorphic]] to its image in <math>X</math> (also with the subspace topology) and <math>\iota</math> is called a [[topological embedding]].
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More generally, suppose  <math>\iota</math> is an injection from a set <math>S</math> to a topological space <math>X</math>. Then the subspace topology on <math>S</math> is defined as the coarsest topology for which <math>\iota</math> is continuous. The open sets in this topology are precisely the ones of the form <math>\iota^{-1}(U)</math> for <math>U</math> open in <math>X</math>. <math>S</math> is then homeomorphic to its image in <math>X</math> (also with the subspace topology) and <math>\iota</math> is called a topological embedding.
  
A subspace <math>S</math> is called an '''open subspace''' if the injection <math>\iota</math> is an [[open map]], i.e., if the forward image of an open set of <math>S</math> is open in <math>X</math>. Likewise it is called a '''closed subspace''' if the injection <math>\iota</math> is a [[closed map]].
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A subspace <math>S</math> is called an '''open subspace''' if the injection <math>\iota</math> is an open map, i.e., if the forward image of an open set of <math>S</math> is open in <math>X</math>. Likewise it is called a '''closed subspace''' if the injection <math>\iota</math> is a closed map.
  
 
== Terminology ==
 
== Terminology ==
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== Examples ==
 
== Examples ==
In the following, <math>\mathbb{R}</math> represents the [[real number]]s with their usual topology.
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In the following, <math>\mathbb{R}</math> represents the real numbers with their usual topology.
* The subspace topology of the [[natural number]]s, as a subspace of <math>\mathbb{R}</math>, is the [[discrete topology]].
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* The subspace topology of the natural numbers, as a subspace of <math>\mathbb{R}</math>, is the discrete topology.
* The [[rational number]]s <math>\mathbb{Q}</math> considered as a subspace of <math>\mathbb{R}</math> do not have the discrete topology ({0} for example is not an open set in <math>\mathbb{Q}</math>). If ''a'' and ''b'' are rational, then the intervals (''a'', ''b'') and [''a'', ''b''] are respectively open and closed, but if ''a'' and ''b'' are irrational, then the set of all rational ''x'' with ''a'' < ''x'' < ''b'' is both open and closed.
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* The rational numbers <math>\mathbb{Q}</math> considered as a subspace of <math>\mathbb{R}</math> do not have the discrete topology ({0} for example is not an open set in <math>\mathbb{Q}</math>). If ''a'' and ''b'' are rational, then the intervals (''a'', ''b'') and [''a'', ''b''] are respectively open and closed, but if ''a'' and ''b'' are irrational, then the set of all rational ''x'' with ''a'' < ''x'' < ''b'' is both open and closed.
 
* The set [0,1] as a subspace of <math>\mathbb{R}</math> is both open and closed, whereas as a subset of <math>\mathbb{R}</math> it is only closed.
 
* The set [0,1] as a subspace of <math>\mathbb{R}</math> is both open and closed, whereas as a subset of <math>\mathbb{R}</math> it is only closed.
* As a subspace of <math>\mathbb{R}</math>, [0, 1] &cup; [2, 3] is composed of two disjoint ''open'' subsets (which happen also to be closed), and is therefore a [[disconnected space]].
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* As a subspace of <math>\mathbb{R}</math>, [0, 1] &cup; [2, 3] is composed of two disjoint ''open'' subsets (which happen also to be closed), and is therefore a disconnected space.
* Let ''S'' = [0, 1) be a subspace of the real line <math>\mathbb{R}</math>. Then [0, {{frac|1|2}}) is open in ''S'' but not in <math>\mathbb{R}</math>. Likewise [{{frac|1|2}}, 1) is closed in ''S'' but not in <math>\mathbb{R}</math>. ''S'' is both open and closed as a subset of itself but not as a subset of <math>\mathbb{R}</math>.
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* Let ''S'' = [0, 1) be a subspace of the real line <math>\mathbb{R}</math>. Then [0, <math> tfrac{1}{2}</math>) is open in ''S'' but not in <math>\mathbb{R}</math>. Likewise [<math> tfrac{1}{2}</math>, 1) is closed in ''S'' but not in <math>\mathbb{R}</math>. ''S'' is both open and closed as a subset of itself but not as a subset of <math>\mathbb{R}</math>.
  
 
== Properties ==
 
== Properties ==
  
The subspace topology has the following characteristic property. Let <math>Y</math> be a subspace of <math>X</math> and let <math>i : Y \to X</math> be the inclusion map. Then for any topological space <math>Z</math> a map <math>f : Z\to Y</math> is continuous [[if and only if]] the composite map <math>i\circ f</math> is continuous.  
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The subspace topology has the following characteristic property. Let <math>Y</math> be a subspace of <math>X</math> and let <math>i : Y \to X</math> be the inclusion map. Then for any topological space <math>Z</math> a map <math>f : Z\to Y</math> is continuous if and only if the composite map <math>i\circ f</math> is continuous.  
 
[[Image:Subspace-01.png|center|Characteristic property of the subspace topology]]
 
[[Image:Subspace-01.png|center|Characteristic property of the subspace topology]]
 
This property is characteristic in the sense that it can be used to define the subspace topology on <math>Y</math>.
 
This property is characteristic in the sense that it can be used to define the subspace topology on <math>Y</math>.
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* Suppose <math>S</math> is an open subspace of <math>X</math> (so <math>S\in\tau</math>). Then a subset of <math>S</math> is open in <math>S</math> if and only if it is open in <math>X</math>.
 
* Suppose <math>S</math> is an open subspace of <math>X</math> (so <math>S\in\tau</math>). Then a subset of <math>S</math> is open in <math>S</math> if and only if it is open in <math>X</math>.
 
* Suppose <math>S</math> is a closed subspace of <math>X</math> (so <math>X\setminus S\in\tau</math>). Then a subset of <math>S</math> is closed in <math>S</math> if and only if it is closed in <math>X</math>.
 
* Suppose <math>S</math> is a closed subspace of <math>X</math> (so <math>X\setminus S\in\tau</math>). Then a subset of <math>S</math> is closed in <math>S</math> if and only if it is closed in <math>X</math>.
* If <math>B</math> is a [[basis (topology)|basis]] for <math>X</math> then <math>B_S = \{U\cap S : U \in B\}</math> is a basis for <math>S</math>.
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* If <math>B</math> is a basis for <math>X</math> then <math>B_S = \{U\cap S : U \in B\}</math> is a basis for <math>S</math>.
* The topology induced on a subset of a [[metric space]] by restricting the [[metric (mathematics)|metric]] to this subset coincides with subspace topology for this subset.
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* The topology induced on a subset of a metric space by restricting the metric to this subset coincides with subspace topology for this subset.
  
 
== Preservation of topological properties ==
 
== Preservation of topological properties ==
  
If a topological space having some [[topological property]] implies its subspaces have that property, then we say the property is '''hereditary'''. If only closed subspaces must share the property we call it '''weakly hereditary'''.
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If a topological space having some topological property implies its subspaces have that property, then we say the property is '''hereditary'''. If only closed subspaces must share the property we call it '''weakly hereditary'''.
 +
 
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* Every open and every closed subspace of a completely metrizable space is completely metrizable.
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* Every open subspace of a Baire space is a Baire space.
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* Every closed subspace of a compact space is compact.
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* Being a Hausdorff space is hereditary.
 +
* Being a normal space is weakly hereditary.
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* Total boundedness is hereditary.
 +
* Being totally disconnected is hereditary.
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* First countability and second countability are hereditary.
  
* Every open and every closed subspace of a [[completely metrizable]] space is completely metrizable.
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= Metric space aimed at its subspace =
* Every open subspace of a [[Baire space]] is a Baire space.
 
* Every closed subspace of a [[compact space]] is compact.
 
* Being a [[Hausdorff space]] is hereditary.
 
* Being a [[normal space]] is weakly hereditary.
 
* [[Total boundedness]] is hereditary.
 
* Being [[totally disconnected]] is hereditary.
 
* [[First countability]] and [[second countability]] are hereditary.
 
  
{{Short description|Universal property of metric spaces}}In [[mathematics]], a '''metric space aimed at its subspace''' is a [[category theory|categorical]] construction that has a direct geometric meaning.  It is also a useful step toward the construction of the ''metric envelope'', or [[tight span]], which are basic (injective) objects of the category of [[metric space]]s.
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In mathematics, a '''metric space aimed at its subspace''' is a categorical construction that has a direct geometric meaning.  It is also a useful step toward the construction of the ''metric envelope'', or tight span, which are basic (injective) objects of the category of metric spaces.
  
Following {{harv|Holsztyński|1966}}, a notion of a metric space ''Y'' aimed at its subspace ''X'' is defined.
+
Following (Holsztyński 1966), a notion of a metric space ''Y'' aimed at its subspace ''X'' is defined.
  
 
== Informal introduction ==
 
== Informal introduction ==
 
Informally, imagine terrain ''Y'', and its part ''X'', such that wherever in ''Y'' you place a sharpshooter, and an apple at another place in ''Y'', and then let the sharpshooter fire, the bullet will go through the apple and will always hit a point of ''X'', or at least it will fly arbitrarily close to points of ''X'' – then we say that ''Y'' is aimed at ''X''.
 
Informally, imagine terrain ''Y'', and its part ''X'', such that wherever in ''Y'' you place a sharpshooter, and an apple at another place in ''Y'', and then let the sharpshooter fire, the bullet will go through the apple and will always hit a point of ''X'', or at least it will fly arbitrarily close to points of ''X'' – then we say that ''Y'' is aimed at ''X''.
  
A priori, it may seem plausible that for a given ''X'' the superspaces ''Y'' that aim at ''X'' can be arbitrarily large or at least huge.  We will see that this is not the case. Among the spaces which aim at a subspace isometric to ''X'', there is a unique ([[up to]] [[isometry]]) [[Universal property|universal]] one, Aim(''X''), which in a sense of canonical [[isometric embedding]]s contains any other space aimed at (an isometric image of) ''X''. And in the special case of an arbitrary compact metric space ''X'' every bounded subspace of an arbitrary metric space ''Y'' aimed at ''X'' is [[totally bounded]] (i.e. its metric completion is compact).
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A priori, it may seem plausible that for a given ''X'' the superspaces ''Y'' that aim at ''X'' can be arbitrarily large or at least huge.  We will see that this is not the case. Among the spaces which aim at a subspace isometric to ''X'', there is a unique (up to isometry) universal one, Aim(''X''), which in a sense of canonical isometric embeddings contains any other space aimed at (an isometric image of) ''X''. And in the special case of an arbitrary compact metric space ''X'' every bounded subspace of an arbitrary metric space ''Y'' aimed at ''X'' is totally bounded (i.e. its metric completion is compact).
  
 
== Definitions ==
 
== Definitions ==
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:<math>|d(p,y) - d(p,z)| > d(y,z) - \epsilon.</math>
 
:<math>|d(p,y) - d(p,z)| > d(y,z) - \epsilon.</math>
  
Let <math>\text{Met}(X)</math> be the space of all real valued [[metric map]]s (non-[[Contraction mapping|contractive]]) of <math>X</math>. Define
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Let <math>\text{Met}(X)</math> be the space of all real valued metric maps (non-contractive) of <math>X</math>. Define
  
 
:<math>\text{Aim}(X) := \{f \in \operatorname{Met}(X) : f(p) + f(q) \ge d(p,q) \text{ for all } p,q\in X\}.</math>
 
:<math>\text{Aim}(X) := \{f \in \operatorname{Met}(X) : f(p) + f(q) \ge d(p,q) \text{ for all } p,q\in X\}.</math>
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Thus it follows that every space aimed at ''X'' can be isometrically mapped into Aim(X), with some additional (essential) categorical requirements satisfied.
 
Thus it follows that every space aimed at ''X'' can be isometrically mapped into Aim(X), with some additional (essential) categorical requirements satisfied.
  
The space Aim(X) is [[injective metric space|injective]] (hyperconvex in the sense of [[Aronszajn]]-Panitchpakdi) – given a metric space ''M,'' which contains Aim(X) as a metric subspace, there is a canonical (and explicit) metric retraction of ''M'' onto Aim(X) {{harv|Holsztyński|1966}}.
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The space Aim(X) is injective (hyperconvex in the sense of Aronszajn-Panitchpakdi) – given a metric space ''M,'' which contains Aim(X) as a metric subspace, there is a canonical (and explicit) metric retraction of ''M'' onto Aim(X) (Holsztyński 1966).
  
== Licensing ==  
+
= Licensing =
 
Content obtained and/or adapted from:
 
Content obtained and/or adapted from:
 
* [https://en.wikipedia.org/wiki/Subspace_topology Subspace topology, Wikipedia] under a CC BY-SA license
 
* [https://en.wikipedia.org/wiki/Subspace_topology Subspace topology, Wikipedia] under a CC BY-SA license
 
* [https://en.wikipedia.org/wiki/Metric_space_aimed_at_its_subspace Metric space aimed at its subspace, Wikipedia] under a CC BY-SA license
 
* [https://en.wikipedia.org/wiki/Metric_space_aimed_at_its_subspace Metric space aimed at its subspace, Wikipedia] under a CC BY-SA license

Latest revision as of 14:30, 23 January 2022

Subspace topology

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology (or the relative topology, or the induced topology, or the trace topology).

Definition

Given a topological space and a subset of , the subspace topology on is defined by

That is, a subset of is open in the subspace topology if and only if it is the intersection of with an open set in . If is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of . Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated.

Alternatively we can define the subspace topology for a subset of as the coarsest topology for which the inclusion map

is continuous.

More generally, suppose is an injection from a set to a topological space . Then the subspace topology on is defined as the coarsest topology for which is continuous. The open sets in this topology are precisely the ones of the form for open in . is then homeomorphic to its image in (also with the subspace topology) and is called a topological embedding.

A subspace is called an open subspace if the injection is an open map, i.e., if the forward image of an open set of is open in . Likewise it is called a closed subspace if the injection is a closed map.

Terminology

The distinction between a set and a topological space is often blurred notationally, for convenience, which can be a source of confusion when one first encounters these definitions. Thus, whenever is a subset of , and is a topological space, then the unadorned symbols "" and "" can often be used to refer both to and considered as two subsets of , and also to and as the topological spaces, related as discussed above. So phrases such as " an open subspace of " are used to mean that is an open subspace of , in the sense used below; that is: (i) ; and (ii) is considered to be endowed with the subspace topology.

Examples

In the following, represents the real numbers with their usual topology.

  • The subspace topology of the natural numbers, as a subspace of , is the discrete topology.
  • The rational numbers considered as a subspace of do not have the discrete topology ({0} for example is not an open set in ). If a and b are rational, then the intervals (a, b) and [a, b] are respectively open and closed, but if a and b are irrational, then the set of all rational x with a < x < b is both open and closed.
  • The set [0,1] as a subspace of is both open and closed, whereas as a subset of it is only closed.
  • As a subspace of , [0, 1] ∪ [2, 3] is composed of two disjoint open subsets (which happen also to be closed), and is therefore a disconnected space.
  • Let S = [0, 1) be a subspace of the real line . Then [0, ) is open in S but not in . Likewise [, 1) is closed in S but not in . S is both open and closed as a subset of itself but not as a subset of .

Properties

The subspace topology has the following characteristic property. Let be a subspace of and let be the inclusion map. Then for any topological space a map is continuous if and only if the composite map is continuous.

Characteristic property of the subspace topology

This property is characteristic in the sense that it can be used to define the subspace topology on .

We list some further properties of the subspace topology. In the following let be a subspace of .

  • If is continuous then the restriction to is continuous.
  • If is continuous then is continuous.
  • The closed sets in are precisely the intersections of with closed sets in .
  • If is a subspace of then is also a subspace of with the same topology. In other words the subspace topology that inherits from is the same as the one it inherits from .
  • Suppose is an open subspace of (so ). Then a subset of is open in if and only if it is open in .
  • Suppose is a closed subspace of (so ). Then a subset of is closed in if and only if it is closed in .
  • If is a basis for then is a basis for .
  • The topology induced on a subset of a metric space by restricting the metric to this subset coincides with subspace topology for this subset.

Preservation of topological properties

If a topological space having some topological property implies its subspaces have that property, then we say the property is hereditary. If only closed subspaces must share the property we call it weakly hereditary.

  • Every open and every closed subspace of a completely metrizable space is completely metrizable.
  • Every open subspace of a Baire space is a Baire space.
  • Every closed subspace of a compact space is compact.
  • Being a Hausdorff space is hereditary.
  • Being a normal space is weakly hereditary.
  • Total boundedness is hereditary.
  • Being totally disconnected is hereditary.
  • First countability and second countability are hereditary.

Metric space aimed at its subspace

In mathematics, a metric space aimed at its subspace is a categorical construction that has a direct geometric meaning. It is also a useful step toward the construction of the metric envelope, or tight span, which are basic (injective) objects of the category of metric spaces.

Following (Holsztyński 1966), a notion of a metric space Y aimed at its subspace X is defined.

Informal introduction

Informally, imagine terrain Y, and its part X, such that wherever in Y you place a sharpshooter, and an apple at another place in Y, and then let the sharpshooter fire, the bullet will go through the apple and will always hit a point of X, or at least it will fly arbitrarily close to points of X – then we say that Y is aimed at X.

A priori, it may seem plausible that for a given X the superspaces Y that aim at X can be arbitrarily large or at least huge. We will see that this is not the case. Among the spaces which aim at a subspace isometric to X, there is a unique (up to isometry) universal one, Aim(X), which in a sense of canonical isometric embeddings contains any other space aimed at (an isometric image of) X. And in the special case of an arbitrary compact metric space X every bounded subspace of an arbitrary metric space Y aimed at X is totally bounded (i.e. its metric completion is compact).

Definitions

Let be a metric space. Let be a subset of , so that (the set with the metric from restricted to ) is a metric subspace of . Then

Definition.  Space aims at if and only if, for all points of , and for every real , there exists a point of such that

Let be the space of all real valued metric maps (non-contractive) of . Define

Then

for every is a metric on . Furthermore, , where , is an isometric embedding of into ; this is essentially a generalisation of the Kuratowski-Wojdysławski embedding of bounded metric spaces into , where we here consider arbitrary metric spaces (bounded or unbounded). It is clear that the space is aimed at .

Properties

Let be an isometric embedding. Then there exists a natural metric map such that :

for every and .

Theorem The space Y above is aimed at subspace X if and only if the natural mapping is an isometric embedding.

Thus it follows that every space aimed at X can be isometrically mapped into Aim(X), with some additional (essential) categorical requirements satisfied.

The space Aim(X) is injective (hyperconvex in the sense of Aronszajn-Panitchpakdi) – given a metric space M, which contains Aim(X) as a metric subspace, there is a canonical (and explicit) metric retraction of M onto Aim(X) (Holsztyński 1966).

Licensing

Content obtained and/or adapted from: