View source for Metric (mathematics)

{{distinguish|Measure (mathematics)}}
[[File:Manhattan distance.svg|thumb|200px|An illustration comparing the [[Taxicab geometry|taxicab metric]] to the Euclidean metric on the plane: According to the taxicab metric the red, yellow, and blue paths have the same [[Arc length|length]] (12). According to the Euclidean metric, the green path has length <math>6 \sqrt{2} \approx 8.49</math>, and is the unique shortest path.]]

In [[mathematics]], a '''metric''' or '''distance function''' is a [[function (mathematics)|function]] that defines a [[distance]]  between each pair of elements of a [[Set (mathematics)|set]]. A set with a metric is called a [[metric space]].<ref>{{cite book |first=Eduard |last=Čech |authorlink=Eduard Čech |title=Point Sets |location=New York |publisher=Academic Press |year=1969 |page=42 }}</ref> A metric induces a [[topology]] on a set, but not all topologies can be generated by a metric. A [[topological space]] whose topology can be described by a metric is called [[metrizable]].

An important source of metrics in [[differential geometry]] are [[metric tensor]]s, [[bilinear form]]s that may be defined from the [[tangent vector]]s of a [[differentiable manifold]] onto a scalar. A metric tensor allows distances along curves to be determined through integration, and thus determines a metric. However, not every metric comes from a metric tensor in this way.

==Definition==

A '''metric''' on a set {{mvar|X}} is a [[function (mathematics)|function]] (called the ''distance function'' or simply '''distance''')
:<math>d : X \times X \to [0,\infty)</math>,
where <math>[0,\infty)</math> is the set of non-negative [[real number]]s and for all <math>x,y,z \in X</math>, the following conditions are satisfied:

:{|
|-
|style="width:20px"| 1.
|style="width:250px"|<math>d(x,y) \ge 0 </math> 
|| [[Non-negative|non-negativity]] or separation axiom
|-
| 2. || <math>d(x,y) = 0 \Leftrightarrow x = y</math> || [[identity of indiscernibles]]
|-
| 3. || <math>d(x,y)  = d(y,x) </math> || [[Symmetric function|symmetry]]
|-
| 4. || <math>d(x,y) \le d(x,z) + d(z, y)</math>  || [[subadditivity]] or [[triangle inequality]]
|}

Conditions 1 and 2 together define a ''[[positive-definite function]]''. 
The first condition is implied by the others.

A metric is called an [[ultrametric space|ultrametric]] if it satisfies the following stronger version of the ''triangle inequality'' where points can never fall 'between' other points:
: <math>d(x, z) \leq \max(d(x, y), d(y, z))</math>
for all <math>x,y,z \in X</math>

A metric {{mvar|d}} on {{mvar|X}} is called [[intrinsic metric|intrinsic]] if any two points {{mvar|x}} and {{mvar|y}} in {{mvar|X}} can be joined by a [[curve]] with [[Curve#Lengths of curves|length]] arbitrarily close to {{math|''d''(''x'', ''y'')}}.

A metric ''d'' on a group ''G'' (written multiplicatively) is said to be '''left-invariant''' (resp. '''right invariant''') if we have
:<math>d(zx, zy) = d(x, y)</math> [resp. <math>d(xz,yz)=d(x,y)</math>]
for all ''x'', ''y'', and ''z'' in ''G''.

== Notes ==

These conditions express intuitive notions about the concept of [[distance]]. For example, that the distance between distinct points is positive and the distance from ''x'' to ''y'' is the same as the distance from ''y'' to ''x''. The triangle inequality means that the distance from ''x'' to ''z'' via ''y'' is at least as great as from ''x'' to ''z'' directly. [[Euclid]] in his [[Euclidean geometry|work]] stated that the shortest distance between two points is a line; that was the triangle inequality for his geometry.

== Examples ==

{{Main|Metric space#Examples of metric spaces}}
* The [[discrete space|discrete metric]]: if ''x'' = ''y'' then ''d''(''x'',''y'') = 0.  Otherwise, ''d''(''x'',''y'') = 1.
* The [[Euclidean metric]] is translation and rotation invariant.
* The [[Taxicab geometry|taxicab metric]] is translation invariant.
* More generally, any metric induced by a [[norm (mathematics)|norm]] is translation invariant.
* If <math>(p_n)_{n\in N}</math> is a [[sequence]] of [[seminorm]]s defining a ([[locally convex]]) [[topological vector space]] ''E'', then
:<math>d(x,y)=\sum_{n=1}^\infty \frac{1}{2^n} \frac{p_n(x-y)}{1+p_n(x-y)}</math>
:is a metric defining the same [[topology]].  (One can replace <math> \frac{1}{2^n}</math> by any [[absolute convergence|summable sequence]] <math>(a_n)</math> of strictly [[positive number]]s.)
* [[Graph metric]], a metric defined in terms of distances in a certain graph.
* The [[Hamming distance]] in coding theory.
* [[Riemannian metric]], a type of metric function that is appropriate to impose on any [[differentiable manifold]]. For any such [[manifold]], one chooses at each point p a symmetric, positive definite, bilinear form L: T<sub>p</sub> × T<sub>p</sub> → ℝ on the [[tangent space]] T<sub>p</sub> at p, doing so in a smooth manner. This form determines the length of any tangent vector '''v''' on the manifold, via the definition ||v||  =  <math>\sqrt{L(\mathbf{v}, \mathbf{v})}</math>. Then for any differentiable path on the manifold, its length is defined as the integral of the length of the tangent vector to the path at any point, where the integration is done with respect to the path parameter.  Finally, to get a metric defined on any pair {x, y} of points of the manifold, one takes the infimum, over all paths from x to y, of the set of path lengths. A smooth manifold equipped with a Riemannian metric is called a [[Riemannian manifold]].
* The [[Fubini–Study metric]] on [[complex projective space]]. This is an example of a Riemannian metric.
* [[String metric]]s, such as [[Levenshtein distance]] and other [[Edit distance|string edit distances]], define a metric over [[String (computer science)|strings]].
* [[Graph edit distance]] defines a distance function between [[Graph (discrete mathematics)|graphs]].
* The [[Wasserstein metric]] is a distance function defined between two [[probability distribution]]s.
* The [[Finsler metric]] is a continuous nonnegative function F:TM→[0,+∞) defined on the tangent bundle.

== Equivalence of metrics ==

For a given set ''X'', two metrics ''d''<sub>1</sub> and ''d''<sub>2</sub> are called '''topologically equivalent''' ('''uniformly equivalent''') if the identity mapping
:id: (''X'',''d''<sub>1</sub>) → (''X'',''d''<sub>2</sub>)
is a [[homeomorphism]] ([[uniform isomorphism]]).

For example, if <math>d</math> is a metric, then <math>\min (d, 1)</math> and <math>{d \over 1+d}</math> are metrics equivalent to <math>d.</math>

See also [[Metric space#Notions of metric space equivalence|notions of metric space equivalence]].

== Metrics on vector spaces ==
<!-- linked from [[Relation of norms and metrics]] -->
Norms on vector spaces are equivalent to certain metrics, namely homogeneous, translation-invariant ones. In other words, every norm determines a metric, and some metrics determine a norm.

Given a [[normed vector space]] <math>(X, \|\cdot\|)</math> we can define a metric on ''X'' by
:<math>d(x,y) := \| x-y\|</math>.
The metric ''d'' is said to be '''induced by''' the norm <math>\|\cdot\|</math>.

Conversely if a metric ''d'' on a [[vector space]] ''X'' satisfies the properties
*<math>d(x,y) = d(x+a,y+a)</math> (''translation invariance'')
*<math>d(\alpha x, \alpha y) = |\alpha| d(x,y)</math> (''homogeneity'')
then we can define a [[norm (mathematics)|norm]] on ''X'' by
:<math>\|x\| := d(x,0)</math>

Similarly, a [[seminorm]] induces a pseudometric (see below), and a homogeneous, translation invariant pseudometric induces a seminorm.

==Metrics on multisets==
We can generalize the notion of a metric from a distance between two elements to a distance between two nonempty finite multisets of elements. A [[multiset]] is a generalization of the notion of a [[set (mathematics)|set]] such that an element can occur more than once.  Define <math>Z=XY</math> if
<math>Z</math> is the multiset consisting of the elements of the multisets <math>X</math> and <math>Y</math>, that is, if <math>x</math> occurs once in <math>X</math> and once in <math>Y</math> then it occurs twice in <math>Z</math>. 
A distance function
<math>d</math> on the set of nonempty finite multisets is a metric<ref name="Vi11">{{cite journal|doi=10.1109/TIT.2011.2110130|title=Information Distance in Multiples|journal=IEEE Transactions on Information Theory|volume=57|issue=4|pages=2451|year=2011|last1=Vitanyi|first1=Paul M. B.|arxiv=0905.3347}}</ref> if
#  <math>d(X)=0</math> if all elements of <math>X</math> are equal and <math>d(X) > 0</math> otherwise ([[positive definiteness]]), that is, ([[Non-negative|non-negativity]] plus [[identity of indiscernibles]])
# <math>d(X)</math> is invariant under all permutations of <math>X</math> ([[symmetry]])
# <math>d(XY) \leq d(XZ)+d(ZY)</math> ([[triangle inequality]])
Note that the familiar metric between two elements results if the multiset <math>X</math> has two elements in 1 and 2 and the multisets <math>X,Y,Z</math> have one element each in 3. For instance if <math>X</math> consists of two occurrences of <math>x</math>, then <math>d(X)=0</math> according to 1.

A simple example is the set of all nonempty finite multisets <math>X</math>  of integers with <math>d(X)=\max\{x: x \in X\}- \min\{x:x \in X\}</math>. More complex examples are [[information distance]] in multisets;<ref name="Vi11"/> and [[normalized compression distance]] (NCD) in multisets.<ref>{{Cite journal|arxiv=1212.5711|last1= Cohen|first1= Andrew R.|title= Normalized Compression Distance of Multisets with Applications|journal= IEEE Transactions on Pattern Analysis and Machine Intelligence|volume= 37|issue= 8|pages= 1602–1614|last2=  Vitanyi|first2= Paul M. B.|year= 2012|doi= 10.1109/TPAMI.2014.2375175}}
</ref>

== Generalized metrics ==
There are numerous ways of relaxing the axioms of metrics, giving rise to various notions of generalized metric spaces. These generalizations can also be combined. The terminology used to describe them is not completely standardized. Most notably, in [[functional analysis]] pseudometrics often come from [[seminorm]]s on vector spaces, and so it is natural to call them "semimetrics". This conflicts with the use of the term in [[topology]].

=== Extended metrics ===
Some authors allow the distance function ''d'' to attain the value ∞, i.e. distances are non-negative numbers on the [[extended real number line]]. 
Such a function is called an ''extended metric'' or "∞-metric". 
Every extended metric can be transformed to a finite metric such that the metric spaces are equivalent as far as notions of [[topology]] (such as [[continuous function (topology)|continuity]] or [[limit (mathematics)|convergence]]) are concerned. This can be done using a [[Subadditive function|subadditive]] monotonically increasing bounded function which is zero at zero, e.g. ''d''&prime;(''x'', ''y'') = ''d''(''x'', ''y'') / (1 + ''d''(''x'', ''y'')) or ''d''&prime;&prime;(''x'', ''y'') = min(1, ''d''(''x'', ''y'')).

The requirement that the metric take values in <nowiki>[0,∞)</nowiki> can even be relaxed to consider metrics with values in other [[directed set]]s.  The reformulation of the axioms in this case leads to the construction of [[uniform space]]s: topological spaces with an abstract structure enabling one to compare the local topologies of different points.

===Pseudometrics===
{{Main|Pseudometric space}}
A '''pseudometric''' on ''X'' is a function ''d'' : ''X'' × ''X'' → '''R''' which satisfies the axioms for a metric, except that instead of the second (identity of indiscernibles) only ''d''(''x'',''x'')=0 for all ''x'' is required. In other words, the axioms for a pseudometric are:

# ''d''(''x'', ''y'') ≥ 0
<!--# ''d''(''x'', ''y'') = 0   if and only if   ''x'' = ''y''-->
# '''''d''(''x'', ''x'') = 0''' (but possibly ''d''(''x'', ''y'') = 0 for some distinct values ''x'' ≠ ''y''.)
# ''d''(''x'', ''y'') = ''d''(''y'', ''x'')
# ''d''(''x'', ''z'') ≤ ''d''(''x'', ''y'') + ''d''(''y'', ''z'').

In some contexts, pseudometrics are referred to as ''semimetrics'' because of their relation to [[seminorm]]s.

===Quasimetrics===
Occasionally, a '''quasimetric''' is defined as a function that satisfies all axioms for a metric with the possible exception of symmetry:<ref>E.g. Steen &amp; Seebach (1995).</ref><ref>{{cite conference | title=Quasi uniformities: reconciling domains with metric spaces | last=Smyth | first=M. | year=1987 | pages=236–253 | conference=3rd Conference on Mathematical Foundations of Programming Language Semantics | editor=M.Main |editor2=A.Melton |editor3=M.Mislove |editor4=D.Schmidt | publisher=Springer-Verlag, Lecture Notes in Computer Science 298|url=https://link.springer.com/chapter/10.1007/3-540-19020-1_12}}</ref>. The name of this generalisation is not entirely standardized.<ref>{{Citation | title=Functional Analysis and Control Theory: Linear Systems | last=Rolewicz | first=Stefan | year=1987 | isbn=90-277-2186-6 | publisher=[[Springer Science+Business Media|Springer]] | oclc=13064804 }} This book calls them "semimetrics". That same term is also frequently used for two other generalizations of metrics.</ref>

# ''d''(''x'', ''y'') ≥ 0 ('''positivity''')
# ''d''(''x'', ''y'') = 0 &nbsp; if and only if &nbsp; ''x'' = ''y'' ('''positive definiteness''')
# <del>''d''(''x'', ''y'') = ''d''(''y'', ''x'')</del> ('''symmetry''', dropped)
# ''d''(''x'', ''z'') ≤ ''d''(''x'', ''y'') + ''d''(''y'', ''z'') ('''triangle inequality''')

Quasimetrics are common in real life. For example, given a set ''X'' of mountain villages, the typical walking times between elements of ''X'' form a quasimetric because travel up hill takes longer than travel down hill. Another example is a [[taxicab geometry]] topology having one-way streets, where a path from point ''A'' to point ''B'' comprises a different set of streets than a path from ''B'' to ''A''. 

A quasimetric on the reals can be defined by setting
:''d''(''x'', ''y'') = ''x'' − ''y'' if ''x'' ≥ ''y'', and
:''d''(''x'', ''y'') = 1 otherwise. The 1 may be replaced by infinity or by <math>1+10^{(y-x)}</math>.
The topological space underlying this quasimetric space is the [[Sorgenfrey line]]. This space describes the process of [[Filing (metalworking)|filing down]] a metal stick: it is easy to reduce its size, but it is difficult or impossible to grow it.

If ''d'' is a quasimetric on ''X'', a metric ''d<nowiki>'</nowiki>'' on ''X'' can be formed by taking
:''d<nowiki>'</nowiki>''(''x'', ''y'') = {{frac|2}}(''d''(''x'', ''y'') + ''d''(''y'', ''x'')).

===Metametrics===

In a ''metametric'', all the axioms of a metric are satisfied except that the distance between identical points is not necessarily zero. In other words, the axioms for a metametric are:

# ''d''(''x'', ''y'') ≥ 0
# ''d''(''x'', ''y'') = 0 implies ''x'' = ''y'' (but not vice versa.)
# ''d''(''x'', ''y'') = ''d''(''y'', ''x'')
# ''d''(''x'', ''z'') ≤ ''d''(''x'', ''y'') + ''d''(''y'', ''z'').

Metametrics appear in the study of [[Δ-hyperbolic space|Gromov hyperbolic metric spaces]] and their boundaries. The ''visual metametric'' on such a space satisfies ''d''(''x'', ''x'') = 0 for points ''x'' on the boundary, but otherwise ''d''(''x'', ''x'') is approximately the distance from ''x'' to the boundary. Metametrics were first defined by Jussi Väisälä.<ref>{{citation
 | last = Väisälä | first = Jussi
 | doi = 10.1016/j.exmath.2005.01.010
 | issue = 3
 | journal = Expositiones Mathematicae
 | mr = 2164775
 | pages = 187–231
 | title = Gromov hyperbolic spaces
 | url = http://www.helsinki.fi/~jvaisala/grobok.pdf
 | volume = 23
 | year = 2005}}</ref>

===Semimetrics===
A '''semimetric''' on ''X'' is a function ''d'' : ''X'' × ''X'' → '''R''' that satisfies the first three axioms, but not necessarily the triangle inequality:

# ''d''(''x'', ''y'') ≥ 0
# ''d''(''x'', ''y'') = 0 &nbsp; if and only if &nbsp; ''x'' = ''y''
# ''d''(''x'', ''y'') = ''d''(''y'', ''x'')

Some authors work with a weaker form of the triangle inequality, such as:
: ''d''(''x'', ''z'') ≤ ρ (''d''(''x'', ''y'') + ''d''(''y'', ''z'')) &nbsp;&nbsp;&nbsp; (ρ-relaxed triangle inequality)
: ''d''(''x'', ''z'') ≤ ρ max(''d''(''x'', ''y''), ''d''(''y'', ''z'')) &nbsp;&nbsp;&nbsp; (ρ-inframetric inequality).
The ρ-inframetric inequality implies the ρ-relaxed triangle inequality (assuming the first axiom), and the ρ-relaxed triangle inequality implies the 2ρ-inframetric inequality. Semimetrics satisfying these equivalent conditions have sometimes been referred to as "quasimetrics",<ref>{{Citation
 | title = The Geodesic Problem in Quasimetric Spaces
 | year = 2009
 | author = Xia, Q.
 | journal = Journal of Geometric Analysis
 | pages = 452–479
 | volume = 19
 | doi = 10.1007/s12220-008-9065-4
 | issue = 2 | arxiv = 0807.3377
 }}</ref> "nearmetrics"<ref>{{Citation | author1=Qinglan Xia | title=The geodesic problem in nearmetric spaces | year=2008 | pages=452–479 | volume=19 | issue=2 | journal=Journal of Geometric Analysis | arxiv=0807.3377 | postscript=.| bibcode=2008arXiv0807.3377X }}</ref> or '''inframetrics'''.<ref name=inframetrics>* {{cite book
 | title = 2008 IEEE INFOCOM - The 27th Conference on Computer Communications
 | citeseerx=10.1.1.113.6748
 | year = 2008
 | journal = IEEE INFOCOM 2008. the 27th Conference on Computer Communications
 | pages = 1085–1093
 | last1 = Fraigniaud	 | first1 =  P.
 | last2 =  Lebhar	 | first2 =  E.
 | last3 =  Viennot	 | first3 =  L.
 | doi = 10.1109/INFOCOM.2008.163
 | chapter = The Inframetric Model for the Internet
 | isbn = 978-1-4244-2026-1 }}.</ref>

The ρ-inframetric inequalities were introduced to model [[round-trip delay time]]s in the [[internet]].<ref name=inframetrics /> The triangle inequality implies the 2-inframetric inequality, and the [[ultrametric inequality]] is exactly the 1-inframetric inequality.

===Premetrics===
Relaxing the last three axioms leads to the notion of a '''premetric''', i.e. a function satisfying the following conditions:

# ''d''(''x'', ''y'') ≥ 0
# ''d''(''x'', ''x'') = ''0''

This is not a standard term. Sometimes it is used to refer to other generalizations of metrics such as pseudosemimetrics<ref>{{Citation
 | title =  Metric characterization of random variables and random processes
 | year = 2000
 | last1 = Buldygin	 | first1 =  V.V.
 | last2 =  Kozachenko	 | first2 =  I.U.V.
 | url = https://books.google.com/books?hl=en&lr=&id=ePDXvIhdEjoC&oi=fnd&pg=PR9&dq=%22Metric+characterization+of+random+variables+and+random+processes%22&ots=NqeEF2xeJl&sig=DDsJXBHPXLApBnVIYHHEPQuLGMk#v=onepage&q=%22Metric%20characterization%20of%20random%20variables%20and%20random%20processes%22&f=false
}}.</ref> or pseudometrics;<ref>{{citation
 | title =  Lectures and exercises on functional analysis
 | year = 2006
 | author = Khelemskiĭ
}}.</ref> in translations of Russian books it sometimes appears as "prametric".<ref>Arkhangel'skii &amp; Pontryagin (1990). {{citation
 | title =  An introduction to geometrical physics
 | year = 1995
 | last1 = Aldrovandi	 | first1 =  R.
 | last2 =  Pereira	 | first2 =  J.G.
}}.</ref>

Any premetric gives rise to a topology as follows. For a positive real ''r'', the ''r''-ball centered at a point ''p'' is defined as
:''B<sub>r</sub>''(''p'') = { ''x'' | ''d''(''x'', ''p'') < r }.
A set is called ''open'' if for any point ''p'' in the set there is an ''r''-ball centered at ''p'' which is contained in the set. Every premetric space is a topological space, and in fact a [[sequential space]].<!--I copied this claim from [[premetric space]] without checking-->
In general, the ''r''-balls themselves need not be open sets with respect to this topology. 
As for metrics, the distance between two sets ''A'' and ''B'', is defined as
:''d''(''A'', ''B'') = inf<sub>''x''∊''A'', ''y''∊''B''</sub> ''d''(''x'', ''y'').
This defines a premetric on the [[power set]] of a premetric space. If we start with a (pseudosemi-)metric space, we get a pseudosemimetric, i.e. a symmetric premetric.
Any premetric gives rise to a [[preclosure operator]] ''cl'' as follows:
:''cl''(''A'') = { ''x'' | ''d''(''x'', ''A'') = 0 }.

===Pseudoquasimetrics===
The prefixes ''pseudo-'', ''quasi-'' and ''semi-'' can also be combined, e.g., a '''pseudoquasimetric''' (sometimes called '''hemimetric''') relaxes both the indiscernibility axiom and the symmetry axiom and is simply a premetric satisfying the triangle inequality. For pseudoquasimetric spaces the open ''r''-balls form a basis of open sets. A very basic example of a pseudoquasimetric space is the set {0,1} with the premetric given by ''d''(0,1) = 1 and ''d''(1,0) = 0. The associated topological space is the [[Sierpiński space]].

Sets equipped with an extended pseudoquasimetric were studied by [[William Lawvere]] as "generalized metric spaces".<ref>{{citation | last=Lawvere | first=F.W. | title=Metric spaces, generalised logic, and closed categories | series=Reprints in Theory and Applications of Categories | volume=1 | year=2002 | origyear=1973 | pages=1–37 |url=http://emis.ams.org/journals/TAC/reprints/articles/1/tr1.pdf}}.</ref><ref>{{citation | last=Vickers | first=Steven | title=Localic completion of generalized metric spaces I | journal=Theory and Applications of Categories | volume=14 | year=2005 | pages=328–356 | url=http://www.tac.mta.ca/tac/volumes/14/15/14-15abs.html }}
</ref> From a [[Category theory|categorical]] point of view, the extended pseudometric spaces and the extended pseudoquasimetric spaces, along with their corresponding nonexpansive maps, are the best behaved of the metric space categories. One can take arbitrary products and coproducts and form quotient objects within the given category. If one drops "extended", one can only take finite products and coproducts. If one drops "pseudo", one cannot take quotients. [[Approach space]]s are a generalization of metric spaces that maintains these good categorical properties.

=== Important cases of generalized metrics ===
In [[differential geometry]], one considers a [[metric tensor]], which can be thought of as an "infinitesimal" quadratic metric function. This is defined as a [[nondegenerate]] symmetric [[bilinear form]] on the [[tangent space]] of a [[manifold]] with an appropriate [[differentiability]] requirement.  While these are not metric functions as defined in this article, they induce what is called a pseudo-semimetric function by [[Antiderivative|integration]] of its square root along a path through the manifold. If one imposes the positive-definiteness requirement of an [[inner product]] on the metric tensor, this restricts to the case of a [[Riemannian manifold]], and the path integration yields a metric.

In [[general relativity]] the related concept is a [[metric tensor (general relativity)]] which expresses the structure of a [[pseudo-Riemannian manifold]]. Though the term "metric" is used, the fundamental idea is different because there are non-zero [[null vector]]s in the tangent space of these manifolds, and vectors can have negative squared norms. This generalized view of "metrics", in which zero distance does ''not'' imply identity, has crept into some mathematical writing too:<ref>S. Parrott (1987) ''Relativistic Electrodynamics and Differential Geometry'', page 4, Springer-Verlag {{isbn|0-387-96435-5}} : "This bilinear form is variously called the ''Lorentz metric'', or ''Minkowski metric'' or ''metric tensor''."</ref><ref>Thomas E. Cecil (1992) ''Lie Sphere Geometry'', page 9, Springer-Verlag {{isbn|0-387-97747-3}} : "We call this scalar product the ''Lorentz metric''"</ref>

== See also ==
*[[Acoustic metric]]
*[[Complete metric]]
*[[Similarity measure]]
*[[Signed distance function]]

== Notes ==
<references />

== References ==
* {{Citation | title=General Topology I: Basic Concepts and Constructions Dimension Theory | last1=Arkhangel'skii | first1=A. V. | last2=Pontryagin |first2=L. S. | year=1990 | isbn=3-540-18178-4 | publisher=[[Springer Science+Business Media|Springer]] | series=Encyclopaedia of Mathematical Sciences}}
*{{citation | last1=Steen | first1=Lynn Arthur | authorlink1=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | authorlink2=J. Arthur Seebach Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=[[Dover Publications|Dover]] | isbn=978-0-486-68735-3 |mr=507446 | year=1995 | oclc=32311847 }}

==External links==
* {{planetmath reference|id=6274|title=Quasimetric space}}
* {{planetmath reference|id=5904|title=Semimetric}}

{{DEFAULTSORT:Metric (Mathematics)}}
[[Category:Metric geometry]]
[[Category:Topology]]