Metric Spaces - Revision history

Khanh: /* Open and closed sets, topology and convergence */

2021-10-27T04:13:14Z

Open and closed sets, topology and convergence

Khanh: /* Open and closed sets, topology and convergence */

2021-10-27T04:12:01Z

Open and closed sets, topology and convergence

Khanh: /* Generalizations of metric spaces */

2021-10-27T04:10:39Z

Generalizations of metric spaces

Khanh: /* Quotient metric spaces */

2021-10-27T04:09:41Z

Quotient metric spaces

Khanh at 04:08, 27 October 2021

2021-10-27T04:08:03Z

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Lila: Created page with "In mathematics, a '''metric space''' is a set together with a metric on the set. The metric is a function (math..."

2021-10-25T20:28:05Z

Created page with "In mathematics, a '''metric space''' is a set together with a metric on the set. The metric is a function (math..."

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

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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>
-These open balls form the [[base (topology)|base]] for a topology on ''M'', making it a topological space.
+These open balls form the base for a topology on ''M'', making it a topological space.
 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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 *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 <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 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.

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 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 if and only if it is a quotient of a metric space.