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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:
* the distance from a point to itself is zero, 
* the distance between two distinct points is positive,   
* the distance from ''A'' to ''B'' is the same as the distance from ''B'' to ''A'', and
* the distance from ''A'' to ''B'' (directly) is less than or equal to the distance from ''A'' to ''B'' via any third point ''C''.

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.

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]].

== History ==
{{Expand section|Reasons for generalizing the Euclidean metric, first non-Euclidean metrics studied, consequences for mathematics|date=August 2011}}

In 1906 [[Maurice Fréchet]] introduced metric spaces in his work ''Sur quelques points du calcul fonctionnel''.<ref>Rendic. Circ. Mat. Palermo 22 (1906) 1–74</ref> However the name is due to [[Felix Hausdorff]].

== Definition ==
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]]

:<math>d \colon M \times M \to \mathbb{R}</math>

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>

<!-- same table is at [[Metric (mathematics)#Definition]] -->
:{|
|-
| 1. || <math>d(x,y) = 0 \Leftrightarrow x = y</math> || [[identity of indiscernibles]]
|-
| 2. || <math>d(x,y)  = d(y,x) </math> || [[Symmetric function|symmetry]]
|-
| 3. || <math>d(x,z) \le d(x,y) + d(y, z)</math>  || [[subadditivity]] or [[triangle inequality]]
|}

Given the above three axioms, we also have that <math>d(x,y) \ge 0 </math> for any <math>x, y \in M</math>. This is deduced as follows:

:{|
|-
|style="width:250px"| <math>d(x,y) + d(y,x) \ge d(x,x)</math>
|| by triangle inequality
|-
| <math>d(x,y) + d(x,y) \ge d(x,x)</math>
| by symmetry
|-
| <math>2d(x,y) \ge 0</math>
| by identity of indiscernibles
|-
| <math>d(x,y) \ge 0</math>
| we have non-negativity
|-
|}

The function <math>d</math> is also called ''distance function'' or simply ''distance''. Often, <math>d</math> is omitted and one just writes <math>M</math> 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 == <!-- [[Finite Metric space]] redirects here-->
* The [[real number]]s with the distance function <math>d(x,y) = \vert y - x \vert</math> given by the [[absolute difference]], and, more generally, [[Euclidean space|Euclidean {{math|''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.
* The [[positive real numbers]] with distance function <math>d(x,y) =\vert \log(y/x) \vert</math> is a complete metric space.
* 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:
** The [[Norm (mathematics)#Taxicab norm or Manhattan norm|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 [[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>.
* 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 . \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.
* 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>.
* 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, any point is an [[open ball]], and therefore every subset is open and the space has the [[discrete topology]].
* 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>
* The [[hyperbolic geometry#Models of the hyperbolic plane|hyperbolic plane]] is a metric space. More generally:
** 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.
** 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 \rightarrow 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.
** 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 <math>y</math>. 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 [[Graph (discrete mathematics)|graphs]], defined as the minimal number of [[Graph operations|graph edit operations]] required to transform one graph into another.
* 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]].
* Given a metric space <math>(X,d)</math> and an increasing [[concave function]] <math>f \colon [0,\infty) \rightarrow [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>.
* 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>.
* 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 <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>.
* 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 <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]].

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.

A topological space which can arise in this way from a metric space is called a '''metrizable''' space; see the article on [[metrization theorems]] for further details.

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> [[iff]] for every <math>\epsilon>0</math>, there exists a natural number ''N'' such that <math>d(x_n,x) < \epsilon </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 iff every sequence in <math>A</math> that converges to a limit in <math>M</math> has its limit in <math>A</math>.

== Types of metric spaces ==

===Complete spaces===
{{Main|Complete metric space}}

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>.

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.

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.

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 [[iff]] it is closed in any containing metric space.

Every complete metric space is a [[Baire space]].

===Bounded and totally bounded spaces===
[[File:Diameter of a Set.svg|thumb|Diameter of a set.]]
{{See also|bounded set}}
A metric space ''M'' is called '''bounded''' if there exists some number ''r'', such that ''d''(''x'',''y'') ≤ ''r'' for all ''x'' and ''y'' in ''M''.  The smallest possible such ''r'' is called the '''[[diameter]]''' of ''M''. The space ''M'' is called '''precompact''' or '''[[totally bounded]]''' if for every ''r'' > 0 there exist finitely many open balls of radius ''r'' whose union covers ''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 [[Interval (mathematics)|interval]]s in the space of [[real number]]s and occasionally regions in a Euclidean space <math>\mathbb 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>\mathbb R ^n</math> may have a finite [[Lebesgue measure|volume]].

===Compact spaces===
A metric space ''M'' is compact if every sequence in ''M'' has a [[subsequence]] that converges to a point in ''M''. 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.

Examples of compact metric spaces include the closed interval [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 [[iff]] 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 ''M'', there exists a "Lebesgue number" δ such that every subset of ''M'' of [[diameter]] < δ 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|accessdate=2 May 2018|url-status=live|archiveurl=https://web.archive.org/web/20090205003114/http://planetmath.org/encyclopedia/CompactMetricSpacesAreSecondCountable.html|archivedate=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]].)

===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 space|proper]] if every closed ball {''y'' : ''d''(''x'',''y'') ≤ ''r''} is compact. Proper spaces are locally compact, but the converse is not true in general.

===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 [[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>.
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]].

[[Simply connected space]]s 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 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.

===Pointed metric spaces===
If <math>X</math> is a nonempty metric space and <math>x_0\in X</math> then <math>(X,x_0)</math> is called a ''pointed metric space'', and <math>x_0</math> 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 <math>0</math> due to its similar behavior to zero in certain contexts.

==Types of maps between metric spaces==
Suppose (''M''<sub>1</sub>,''d''<sub>1</sub>) and (''M''<sub>2</sub>,''d''<sub>2</sub>) are two metric spaces.

===Continuous maps===
{{Main|Continuous function (topology)}}

The map ''f'':''M''<sub>1</sub>→''M''<sub>2</sub> is [[continuous map|continuous]]
if it has one (and therefore all) of the following equivalent properties:
;General topological continuity: for every open set ''U'' in ''M''<sub>2</sub>, the [[preimage]] ''f''<sup>&nbsp;-1</sup>(''U'') is open in ''M''<sub>1</sub>
:This is the general definition of [[Continuity (topology)|continuity in topology]].
;Sequential continuity: if (''x''<sub>''n''</sub>) is a sequence in ''M''<sub>1</sub> that converges to ''x'' in ''M''<sub>1</sub>, then the sequence (''f''(''x''<sub>''n''</sub>)) converges to ''f''(''x'') in ''M''<sub>2</sub>.
:This is [[sequential continuity]], due to [[Eduard Heine]].
;ε-δ definition: for every ''x'' in ''M''<sub>1</sub> and every ε>0 there exists δ>0 such that for all ''y'' in ''M''<sub>1</sub> we have
::<math>d_1(x,y)<\delta \Rightarrow d_2(f(x),f(y))< \varepsilon.</math>
:This uses the [[(ε, δ)-definition of limit]], and is due to [[Augustin Louis Cauchy]].

Moreover, ''f'' is continuous if and only if it is continuous on every compact subset of ''M''<sub>1</sub>.

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.

===Uniformly continuous maps===
The map ''ƒ''&nbsp;: ''M''<sub>1</sub>&nbsp;→ ''M''<sub>2</sub> is [[uniform continuity|uniformly continuous]] if for every ''ε''&nbsp;> 0 there exists ''δ''&nbsp;> 0 such that

:<math>d_1(x,y)<\delta \Rightarrow d_2(f(x),f(y))< \varepsilon \quad\mbox{for all}\quad x,y\in M_1.</math>

Every uniformly continuous map ''ƒ''&nbsp;: ''M''<sub>1</sub>&nbsp;→ ''M''<sub>2</sub> is continuous.  The converse is true if ''M''<sub>1</sub> is compact ([[Heine–Cantor theorem]]).

Uniformly continuous maps turn Cauchy sequences in ''M''<sub>1</sub> into Cauchy sequences in ''M''<sub>2</sub>. For continuous maps this is generally wrong; for example, a continuous map
from the open interval (0,1) ''onto'' the real line turns some Cauchy sequences into unbounded sequences.

===Lipschitz-continuous maps and contractions===
{{Main|Lipschitz continuity}}

Given a real number ''K''&nbsp;> 0, the map ''ƒ''&nbsp;: ''M''<sub>1</sub>&nbsp;→&nbsp;''M''<sub>2</sub> is [[Lipschitz continuity|''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>

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

If ''K'' < 1, then ''ƒ'' is called a [[contraction mapping|contraction]]. Suppose ''M''<sub>2</sub>&nbsp;= ''M''<sub>1</sub> and ''M''<sub>1</sub> is complete. If ''ƒ'' is a contraction, then ''ƒ'' admits a unique fixed point ([[Banach fixed point theorem]]). If ''M''<sub>1</sub> is compact, the condition can be weakened a bit: ''ƒ'' 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>.

===Isometries===
The map ''f'':''M''<sub>1</sub>→''M''<sub>2</sub> 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>
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 ''f''&nbsp;:&nbsp;''M''<sub>1</sub>&nbsp;→&nbsp;''M''<sub>2</sub> is a [[quasi-isometry]] if there exist constants ''A''&nbsp;≥&nbsp;1 and ''B''&nbsp;≥&nbsp;0 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 \text{ for all } x,y\in M_1</math>

and a constant ''C''&nbsp;≥&nbsp;0 such that every point in ''M''<sub>2</sub> has a distance at most ''C'' from some point in the image&nbsp;''f''(''M''<sub>1</sub>).

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 (''M''<sub>1</sub>, ''d''<sub>1</sub>) and (''M''<sub>2</sub>, ''d''<sub>2</sub>):

*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]]<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 [[first countable]] since one can use balls with rational radius as a neighborhood base.

The metric topology on a metric space ''M'' is the coarsest topology on ''M'' relative to which the metric ''d'' is a continuous map from the product of ''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#Distances between sets and between a point and a set|distance between the point and the set]]. If (''M'',''d'') is a metric space, ''S'' is a [[subset]] of ''M'' and ''x'' is a point of ''M'', we define the distance from ''x'' to ''S'' as
:<math>d(x,S) = \inf\{d(x,s) : s \in S \} </math> where <math>\inf</math> represents the [[infimum]].

Then ''d''(''x'', ''S'') = 0 if and only if ''x'' belongs to the [[closure (topology)|closure]] of ''S''. Furthermore, we have the following generalization of the triangle inequality:
:<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.

Given two subsets ''S'' and ''T'' of ''M'', 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]].

In general, the Hausdorff distance ''d''<sub>H</sub>(''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 ''d''<sub>H</sub> turns the set ''K''(''M'') of all non-empty compact subsets of ''M'' into a metric space. One can show that ''K''(''M'') is complete if ''M'' is complete.
(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.

==Product metric spaces==
If <math>(M_1,d_1),\ldots,(M_n,d_n)</math> are metric spaces, and ''N'' is the [[Euclidean norm]] on ''R<sup>n</sup>'', then <math>\Big(M_1\times \ldots \times M_n, N(d_1,\ldots,d_n)\Big)</math> is a metric space, where the [[product metric]] is defined by

:<math>N(d_1,...,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 ''N'' is the [[Norm (mathematics)#Taxicab norm or Manhattan norm|taxicab norm]], a [[Norm (mathematics)#p-norm|p-norm]], the [[Norm (mathematics)#Infinity norm or maximum norm|max norm]], or any other norm which is non-decreasing as the coordinates of a positive ''n''-tuple increase (yielding the triangle inequality).

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

:<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>\mathbf{R}^\mathbf{R}</math> is not [[first-countable space|first-countable]] and thus isn't metrizable.

===Continuity of distance===
In the case of a single space <math>(M,d)</math>, the distance map <math>d\colon M\times M \rightarrow 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>.

==Quotient metric spaces==
If ''M'' is a metric space with metric ''d'', and ''~'' is an [[equivalence relation]] on ''M'', then we can endow the quotient set ''M/~'' with the following (pseudo)metric. Given two equivalence classes [''x''] and [''y''], we define
:<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 nice equivalence relations (e.g., those given by gluing together polyhedra along faces), it is a metric.

The quotient metric ''d'' is characterized by the following [[universal property]]. If <math>f:(M,d)\longrightarrow(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 ''x'', ''y'') satisfying ''f''(''x'')=''f''(''y'') whenever <math>x\sim y,</math> then the induced function <math>\overline{f}\colon M/\sim\longrightarrow X</math>, given by <math>\overline{f}([x])=f(x)</math>, is a metric map <math>\overline{f}\colon (M/\sim,d')\longrightarrow (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>

==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.
*If we consider the first definition of a metric space [[#Definition|given above]] and relax the second requirement, we arrive at 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&printsec=frontcover#v=onepage&q=%22metric%20space%22&f=false|date=19 April 2016|publisher=CRC Press|isbn=978-1-4398-2962-2|author1-link=Pascal Hitzler}}</ref> If we remove the third or fourth, we arrive at a [[quasimetric space]], or a [[semimetric space]].
*If the distance function takes values in the [[extended real number line]] '''R'''∪{+∞}, but otherwise satisfies all four conditions, 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"/>
*[[Approach space]]s 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 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|accessdate=2 May 2018|url-status=live|archiveurl=https://web.archive.org/web/20170727003912/http://www.dcs.warwick.ac.uk/pmetric/|archivedate=27 July 2017}}</ref>

=== 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>.
Every metric space <math>(M,d)</math> can now be viewed as a category <math>M^*</math> [[enriched category|enriched]] over <math>R^*</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>
* 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>.
* Since <math>R^*</math> is a poset, all [[Diagram (category theory)|diagrams]] that are required for an enriched category commute automatically.

See the paper by F.W. Lawvere listed below.

==See also==
*[[Space (mathematics)]]
*[[Metric (mathematics)]]
*[[Metric signature]]
*[[Metric tensor]]
*[[Metric tree]]
*[[Norm (mathematics)]]
*[[Normed vector space]]
*[[Measure (mathematics)]]
*[[Hilbert space]]
*[[Hilbert's fourth problem]]
*[[Product metric]]
*[[Aleksandrov–Rassias problem]]
*[[Category of metric spaces]]
*[[Classical Wiener space]]
*[[Glossary of Riemannian and metric geometry]]
*[[Isometry]], [[contraction mapping]] and [[metric map]]
*[[Lipschitz continuity]]
*[[Triangle inequality]]
*[[Ultrametric space]]

==Notes==
{{Reflist}}

==References==
* Victor Bryant, ''Metric Spaces: Iteration and Application'', [[Cambridge University Press]], 1985, {{isbn|0-521-31897-1}}.
* Dmitri Burago, [[Yuri Dmitrievich Burago|Yu D Burago]], Sergei Ivanov, ''A Course in Metric Geometry'', American Mathematical Society, 2001, {{isbn|0-8218-2129-6}}.
* Athanase Papadopoulos, ''Metric Spaces, Convexity and Nonpositive Curvature'', [[European Mathematical Society]], First edition 2004, {{isbn|978-3-03719-010-4}}. Second edition 2014, {{isbn|978-3-03719-132-3}}. 
* [http://mathsci.ucd.ie/~mos Mícheál Ó Searcóid], [http://mathsci.ucd.ie/~mos/Books/Metric_Spaces ''Metric Spaces''], [https://www.springer.com/1-84628-369-8 Springer Undergraduate Mathematics Series], 2006, {{isbn|1-84628-369-8}}.
* Lawvere, F. William, "Metric spaces, generalized logic, and closed categories", [Rend. Sem. Mat. Fis. Milano 43 (1973), 135—166 (1974); (Italian summary)
This is reprinted (with author commentary) at [http://www.tac.mta.ca/tac/reprints/index.html Reprints in Theory and Applications of Categories]
Also (with an author commentary) in Enriched categories in the logic of geometry and analysis. Repr. Theory Appl. Categ. No. 1 (2002), 1–37.

* {{mathworld|urlname=ProductMetric|title=Product Metric}}

== External links ==
* {{Springer |title=Metric space |id=p/m063680}}
* [http://www.cut-the-knot.org/do_you_know/far_near.shtml Far and near &mdash; several examples of distance functions] at [[cut-the-knot]].

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[[Category:Mathematical structures]]
[[Category:Topology]]
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