Metric space

In 1906 Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel.[1] However the name is due to Felix Hausdorff.

A metric space is an ordered pair ${\displaystyle (M,d)}$ where ${\displaystyle M}$ is a set and ${\displaystyle d}$ is a metric on ${\displaystyle M}$, i.e., a function

${\displaystyle d\,\colon M\times M\to \mathbb {R} }$

such that for any ${\displaystyle x,y,z\in M}$, the following holds:[2]

1.	${\displaystyle d(x,y)=0\iff x=y}$	identity of indiscernibles
2.	${\displaystyle d(x,y)=d(y,x)}$	symmetry
3.	${\displaystyle d(x,z)\leq d(x,y)+d(y,z)}$	subadditivity or triangle inequality

Given the above three axioms, we also have that ${\displaystyle d(x,y)\geq 0}$ for any ${\displaystyle x,y\in M}$. This is deduced as follows:

${\displaystyle d(x,y)+d(y,x)\geq d(x,x)}$	by triangle inequality
${\displaystyle d(x,y)+d(x,y)\geq d(x,x)}$	by symmetry
${\displaystyle 2d(x,y)\geq 0}$	by identity of indiscernibles
${\displaystyle d(x,y)\geq 0}$	we have non-negativity

The function ${\displaystyle d}$ is also called distance function or simply distance. Often, ${\displaystyle d}$ is omitted and one just writes ${\displaystyle M}$ for a metric space if it is clear from the context what metric is used.

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

• The real numbers with the distance function ${\displaystyle d(x,y)=\vert y-x\vert }$ given by the absolute difference, and, more generally, Euclidean n-space with the Euclidean distance, are complete metric spaces. The rational numbers with the same distance function also form a metric space, but not a complete one.
• The positive real numbers with distance function ${\displaystyle d(x,y)=\vert \log(y/x)\vert }$ is a complete metric space.
• Any normed vector space is a metric space by defining ${\displaystyle d(x,y)=\lVert y-x\rVert }$, see also metrics on vector spaces. (If such a space is complete, we call it a Banach space.) Examples:
  • The Manhattan norm gives rise to the Manhattan distance, where the distance between any two points, or vectors, is the sum of the differences between corresponding coordinates.
  • The cyclic Mannheim metric or Mannheim distance is a modulo variant the Manhattan metric.[3][4]
  • The maximum norm gives rise to the Chebyshev distance or chessboard distance, the minimal number of moves a chess king would take to travel from ${\displaystyle x}$ to ${\displaystyle y}$.
• The British Rail metric (also called the “post office metric” or the “SNCF metric”) on a normed vector space is given by ${\displaystyle d(x,y)=\lVert x\rVert +\lVert y\rVert }$ for distinct points ${\displaystyle x}$ and ${\displaystyle y}$, and ${\displaystyle d(x,x)=0}$. More generally ${\displaystyle \lVert .\rVert }$ can be replaced with a function ${\displaystyle f}$ taking an arbitrary set ${\displaystyle S}$ to non-negative reals and taking the value ${\displaystyle 0}$ at most once: then the metric is defined on ${\displaystyle S}$ by ${\displaystyle d(x,y)=f(x)+f(y)}$ for distinct points ${\displaystyle x}$ and ${\displaystyle y}$, and ${\displaystyle d(x,x)=0}$. The name alludes to the tendency of railway journeys to proceed via London (or Paris) irrespective of their final destination.
• If ${\displaystyle (M,d)}$ is a metric space and ${\displaystyle X}$ is a subset of ${\displaystyle M}$, then ${\displaystyle (X,d)}$ becomes a metric space by restricting the domain of ${\displaystyle d}$ to ${\displaystyle X\times X}$.
• The discrete metric, where ${\displaystyle d(x,y)=0}$ if ${\displaystyle x=y}$ and ${\displaystyle d(x,y)=1}$ otherwise, is a simple but important example, and can be applied to all sets. This, in particular, shows that for any set, there is always a metric space associated to it. Using this metric, 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 number of points. Not every finite metric space can be isometrically embedded in a Euclidean space.[5][6]
• The hyperbolic plane is a metric space. More generally:
  • If ${\displaystyle M}$ is any connected Riemannian manifold, then we can turn ${\displaystyle M}$ into a metric space by defining the distance of two points as the infimum of the lengths of the paths (continuously differentiable curves) connecting them.
  • If ${\displaystyle X}$ is some set and ${\displaystyle M}$ is a metric space, then, the set of all bounded functions ${\displaystyle f\colon X\rightarrow M}$ (i.e. those functions whose image is a bounded subset of ${\displaystyle M}$) can be turned into a metric space by defining ${\displaystyle d(f,g)=\sup _{x\in X}d(f(x),g(x))}$ for any two bounded functions ${\displaystyle f}$ and ${\displaystyle g}$ (where ${\displaystyle \sup }$ is supremum).[7] This metric is called the uniform metric or supremum metric, and If ${\displaystyle M}$ is complete, then this function space is complete as well. If X is also a topological space, then the set of all bounded continuous functions from ${\displaystyle X}$ to ${\displaystyle M}$ (endowed with the uniform metric), will also be a complete metric if M is.
  • If ${\displaystyle G}$ is an undirected connected graph, then the set ${\displaystyle V}$ of vertices of ${\displaystyle G}$ can be turned into a metric space by defining ${\displaystyle d(x,y)}$ to be the length of the shortest path connecting the vertices ${\displaystyle x}$ and ${\displaystyle y}$. In geometric group theory this is applied to the Cayley graph of a group, yielding the word metric.
• Graph edit distance is a measure of dissimilarity between two graphs, defined as the minimal number of graph edit operations required to transform one graph into another.
• The Levenshtein distance is a measure of the dissimilarity between two strings ${\displaystyle u}$ and ${\displaystyle v}$, defined as the minimal number of character deletions, insertions, or substitutions required to transform ${\displaystyle u}$ into ${\displaystyle v}$. 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 ${\displaystyle (X,d)}$ and an increasing concave function ${\displaystyle f\colon [0,\infty )\rightarrow [0,\infty )}$ such that ${\displaystyle f(x)=0}$ if and only if ${\displaystyle x=0}$, then ${\displaystyle f\circ d}$ is also a metric on ${\displaystyle X}$.
• Given an injective function ${\displaystyle f}$ from any set ${\displaystyle A}$ to a metric space ${\displaystyle (X,d)}$, ${\displaystyle d(f(x),f(y))}$ defines a metric on ${\displaystyle A}$.
• Using T-theory, the tight span of a metric space is also a metric space. The tight span is useful in several types of analysis.
• The set of all ${\displaystyle m}$ by ${\displaystyle n}$ matrices over some field is a metric space with respect to the rank distance ${\displaystyle d(X,Y)=\mathrm {rank} (Y-X)}$.
• The Helly metric is used in game theory.

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 ${\displaystyle x}$ in a metric space ${\displaystyle M}$ we define the open ball of radius ${\displaystyle r>0}$ (where ${\displaystyle r}$ is a real number) about ${\displaystyle x}$ as the set

${\displaystyle B(x;r)=\{y\in M:d(x,y)<r\}.}$

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

Explicitly, a subset ${\displaystyle U}$ of ${\displaystyle M}$ is called open if for every ${\displaystyle x}$ in ${\displaystyle U}$ there exists an ${\displaystyle r>0}$ such that ${\displaystyle B(x;r)}$ is contained in ${\displaystyle U}$. The complement of an open set is called closed. A neighborhood of the point ${\displaystyle x}$ is any subset of ${\displaystyle M}$ that contains an open ball about ${\displaystyle x}$ as a subset.

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

A sequence (${\displaystyle x_{n}}$) in a metric space ${\displaystyle M}$ is said to converge to the limit ${\displaystyle x\in M}$ if and only if for every ${\displaystyle \varepsilon >0}$, there exists a natural number N such that ${\displaystyle d(x_{n},x)<\varepsilon }$ for all ${\displaystyle n>N}$. Equivalently, one can use the general definition of convergence available in all topological spaces.

A subset ${\displaystyle A}$ of the metric space ${\displaystyle M}$ is closed if and only if every sequence in ${\displaystyle A}$ that converges to a limit in ${\displaystyle M}$ has its limit in ${\displaystyle A}$.

Suppose ${\displaystyle (M_{1},d_{1})}$ and ${\displaystyle (M_{2},d_{2})}$ are two metric spaces.

Given two metric spaces ${\displaystyle (M_{1},d_{1})}$ and ${\displaystyle (M_{2},d_{2})}$:

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

Metric spaces are paracompact[9] Hausdorff spaces[10] and hence normal (indeed they are perfectly normal). An important consequence is that every metric space admits partitions of unity and that every continuous real-valued function defined on a closed subset of a metric space can be extended to a continuous map on the whole space (Tietze extension theorem). It is also true that every real-valued Lipschitz-continuous map defined on a subset of a metric space can be extended to a Lipschitz-continuous map on the whole space.

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

The metric topology on a metric space ${\displaystyle M}$ is the coarsest topology on ${\displaystyle M}$ relative to which the metric ${\displaystyle d}$ is a continuous map from the product of ${\displaystyle M}$ with itself to the non-negative real numbers.

A simple way to construct a function separating a point from a closed set (as required for a completely regular space) is to consider the distance between the point and the set. If ${\displaystyle (M,d)}$ is a metric space, ${\displaystyle S}$ is a subset of ${\displaystyle M}$ and ${\displaystyle x}$ is a point of ${\displaystyle M}$, we define the distance from ${\displaystyle x}$ to ${\displaystyle S}$ as

${\displaystyle d(x,S)=\inf\{d(x,s):s\in S\}}$ where ${\displaystyle \inf }$ represents the infimum.

Then ${\displaystyle d(x,S)=0}$ if and only if ${\displaystyle x}$ belongs to the closure of ${\displaystyle S}$. Furthermore, we have the following generalization of the triangle inequality:

${\displaystyle d(x,S)\leq d(x,y)+d(y,S),}$

which in particular shows that the map ${\displaystyle x\mapsto d(x,S)}$ is continuous.

Given two subsets ${\displaystyle S}$ and ${\displaystyle T}$ of ${\displaystyle M}$, we define their Hausdorff distance to be

${\displaystyle d_{H}(S,T)=\max\{\sup\{d(s,T):s\in S\},\sup\{d(t,S):t\in T\}\}}$ where ${\displaystyle \sup }$ represents the supremum.

In general, the Hausdorff distance ${\displaystyle d_{H}(S,T)}$ can be infinite. Two sets are close to each other in the Hausdorff distance if every element of either set is close to some element of the other set.

The Hausdorff distance ${\displaystyle d_{H}}$ turns the set ${\displaystyle K(M)}$ of all non-empty compact subsets of ${\displaystyle M}$ into a metric space. One can show that ${\displaystyle K(M)}$ is complete if ${\displaystyle M}$ is complete. (A different notion of convergence of compact subsets is given by the Kuratowski convergence.)

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

If ${\displaystyle (M_{1},d_{1}),\ldots ,(M_{n},d_{n})}$ are metric spaces, and ${\displaystyle N}$ is the Euclidean norm on ${\displaystyle \mathbb {R} ^{n}}$, then ${\displaystyle {\Big (}M_{1}\times \ldots \times M_{n},N(d_{1},\ldots ,d_{n}){\Big )}}$ is a metric space, where the product metric is defined by

${\displaystyle N(d_{1},...,d_{n}){\Big (}(x_{1},\ldots ,x_{n}),(y_{1},\ldots ,y_{n}){\Big )}=N{\Big (}d_{1}(x_{1},y_{1}),\ldots ,d_{n}(x_{n},y_{n}){\Big )},}$

and the induced topology agrees with the product topology. By the equivalence of norms in finite dimensions, an equivalent metric is obtained if ${\displaystyle N}$ is the taxicab norm, a p-norm, the maximum norm, or any other norm which is non-decreasing as the coordinates of a positive ${\displaystyle n}$-tuple increase (yielding the triangle inequality).

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

${\displaystyle d(x,y)=\sum _{i=1}^{\infty }{\frac {1}{2^{i}}}{\frac {d_{i}(x_{i},y_{i})}{1+d_{i}(x_{i},y_{i})}}.}$

An uncountable product of metric spaces need not be metrizable. For example, ${\displaystyle \mathbb {R} ^{\mathbb {R} }}$ is not first-countable and thus isn't metrizable.

If M is a metric space with metric ${\displaystyle d}$, and ${\displaystyle \sim }$ is an equivalence relation on ${\displaystyle M}$, then we can endow the quotient set ${\displaystyle M/\!\sim }$ with a pseudometric. Given two equivalence classes ${\displaystyle [x]}$ and ${\displaystyle [y]}$, we define

${\displaystyle d'([x],[y])=\inf\{d(p_{1},q_{1})+d(p_{2},q_{2})+\dotsb +d(p_{n},q_{n})\}}$

where the infimum is taken over all finite sequences ${\displaystyle (p_{1},p_{2},\dots ,p_{n})}$ and ${\displaystyle (q_{1},q_{2},\dots ,q_{n})}$ with ${\displaystyle [p_{1}]=[x]}$, ${\displaystyle [q_{n}]=[y]}$, ${\displaystyle [q_{i}]=[p_{i+1}],i=1,2,\dots ,n-1}$. In general this will only define a pseudometric, i.e. ${\displaystyle d'([x],[y])=0}$ does not necessarily imply that ${\displaystyle [x]=[y]}$. However, for some equivalence relations (e.g., those given by gluing together polyhedra along faces), ${\displaystyle d'}$ is a metric.

The quotient metric ${\displaystyle d}$ is characterized by the following universal property. If ${\displaystyle f\,\colon (M,d)\to (X,\delta )}$ is a metric map between metric spaces (that is, ${\displaystyle \delta (f(x),f(y))\leq d(x,y)}$ for all ${\displaystyle x}$, ${\displaystyle y}$) satisfying ${\displaystyle f(x)=f(y)}$ whenever ${\displaystyle x\sim y,}$ then the induced function ${\displaystyle {\overline {f}}\,\colon M/\!\sim \to X}$, given by ${\displaystyle {\overline {f}}([x])=f(x)}$, is a metric map ${\displaystyle {\overline {f}}\,\colon (M/\!\sim ,d')\to (X,\delta ).}$

A topological space is sequential if and only if it is a quotient of a metric space.[11]

• 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 ${\displaystyle \mathbb {R} \cup \{+\infty \}}$, but otherwise satisfies the conditions of a metric, then it is called an extended metric and the corresponding space is called an ${\displaystyle \infty }$-metric space. If the distance function takes values in some (suitable) ordered set (and the triangle inequality is adjusted accordingly), then we arrive at the notion of generalized ultrametric.[12]
• Approach spaces are a generalization of metric spaces, based on point-to-set distances, instead of point-to-point distances.
• A continuity space is a generalization of metric spaces and posets, that can be used to unify the notions of metric spaces and domains.
• A partial metric space is intended to be the least generalisation of the notion of a metric space, such that the distance of each point from itself is no longer necessarily zero.[13]

• Aleksandrov–Rassias problem
• Category of metric spaces
• Classical Wiener space
• Contraction mapping – Function reducing distance between all points
• Glossary of Riemannian and metric geometry – Mathematics glossary
• Hilbert space – Mathematical generalization to infinite dimension of the notion of Euclidean space
• Hilbert's fourth problem
• Isometry
• Lee distance
• Lipschitz continuity – Strong form of uniform continuity
• Measure (mathematics) – Generalization of length, area, volume and integral
• Metric (mathematics) – Mathematical function defining distance
• Metric map
• Metric signature
• Metric tensor
• Metric tree
• Norm (mathematics) – Length in a vector space
• Normed vector space – Vector space on which a distance is defined
• Product metric
• Space (mathematics) – Mathematical set with some added structure
• Triangle inequality – property of geometry, also used to generalize the notion of "distance" in metric spaces
• Ultrametric space – A type of metric space in which the triangle inequality is replaced by a stronger inequality using max in place of addition

• Victor Bryant, Metric Spaces: Iteration and Application, Cambridge University Press, 1985, ISBN 0-521-31897-1.
• Dmitri 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.
• Mícheál Ó Searcóid, Metric Spaces, 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 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.

• Weisstein, Eric W. "Product Metric". MathWorld.

• "Metric space", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Far and near — several examples of distance functions at cut-the-knot.