Metric space

In mathematics, a metric space is a set together with a metric on the set. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally:

  • the distance from to is zero if and only if and are the same point,
  • the distance between two distinct points is positive,
  • the distance from to is the same as the distance from to , and
  • the distance from to is less than or equal to the distance from to via any third point .

A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces.

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 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. Some of non-geometric metric spaces include spaces of finite strings (finite sequences of symbols from a predefined alphabet) equipped with e.g. a Hamming's or Levenshtein distance, a space of subsets of any metric space equipped with Hausdorff distance, a space of real functions integrable on a unit interval with an integral metric or probabilistic spaces on any chosen metric space equipped with Wasserstein metric. See also the section § Examples of metric spaces.


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   where   is a set and   is a metric on  , i.e., a function


such that for any  , the following holds:[2]

1.   identity of indiscernibles
2.   symmetry
3.   subadditivity or triangle inequality

Given the above three axioms, we also have that   for any  . This is deduced as follows (from the top to the bottom):

  by triangle inequality
  by symmetry
  by identity of indiscernibles
  we have non-negativity

The function   is also called distance function or simply distance. Often,   is omitted and one just writes   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 spacesEdit

  • The real numbers with the distance function   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   is a complete metric space.
  • Any normed vector space is a metric space by defining  , see also metrics on vector spaces. (If such a space is complete, we call it a Banach space.) Examples:
  • The British Rail metric (also called the “post office metric” or the “SNCF metric”) on a normed vector space is given by   for distinct points   and  , and  . More generally   can be replaced with a function   taking an arbitrary set   to non-negative reals and taking the value   at most once: then the metric is defined on   by   for distinct points   and  , and  . The name alludes to the tendency of railway journeys to proceed via London (or Paris) irrespective of their final destination.
  • If   is a metric space and   is a subset of  , then   becomes a metric space by restricting the domain of   to  .
  • The discrete metric, where   if   and   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, the singleton of any point is an open ball, 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   is any connected Riemannian manifold, then we can turn   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   is some set and   is a metric space, then, the set of all bounded functions   (i.e. those functions whose image is a bounded subset of  ) can be turned into a metric space by defining   for any two bounded functions   and   (where   is supremum).[7] This metric is called the uniform metric or supremum metric, and If   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   to   (endowed with the uniform metric), will also be a complete metric if M is.
  • If   is an undirected connected graph, then the set   of vertices of   can be turned into a metric space by defining   to be the length of the shortest path connecting the vertices   and  . 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   and  , defined as the minimal number of character deletions, insertions, or substitutions required to transform   into  . 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   and an increasing concave function   such that   if and only if  , then   is also a metric on  .
  • Given an injective function   from any set   to a metric space  ,   defines a metric on  .
  • 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   by   matrices over some field is a metric space with respect to the rank distance  .
  • The Helly metric is used in game theory.

Open and closed sets, topology and convergenceEdit

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   in a metric space   we define the open ball of radius   (where   is a real number) about   as the set


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

Explicitly, a subset   of   is called open if for every   in   there exists an   such that   is contained in  . The complement of an open set is called closed. A neighborhood of the point   is any subset of   that contains an open ball about   as a subset.

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

A sequence ( ) in a metric space   is said to converge to the limit   if and only if for every  , there exists a natural number N such that   for all  . Equivalently, one can use the general definition of convergence available in all topological spaces.

A subset   of the metric space   is closed if and only if every sequence in   that converges to a limit in   has its limit in  .

Types of metric spacesEdit

Complete spacesEdit

A metric space   is said to be complete if every Cauchy sequence converges in  . That is to say: if   as both   and   independently go to infinity, then there is some   with  .

Every Euclidean space is complete, as is every closed subset of a complete space. The rational numbers, using the absolute value metric  , are not complete.

Every metric space has a unique (up to isometry) completion, which is a complete space that contains the given space as a dense subset. For example, the real numbers are the completion of the rationals.

If   is a complete subset of the metric space  , then   is closed in  . Indeed, a space is complete if and only if it is closed in any containing metric space.

Every complete metric space is a Baire space.

Bounded and totally bounded spacesEdit

Diameter of a set.

A metric space   is called bounded if there exists some number  , such that   for all   The smallest possible such   is called the diameter of   The space   is called precompact or totally bounded if for every   there exist finitely many open balls of radius   whose union covers   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 intervals in the space of real numbers and occasionally regions in a Euclidean space   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   may have a finite volume.

Compact spacesEdit

A metric space   is compact if every sequence in   has a subsequence that converges to a point in  . This is known as sequential compactness and, in metric spaces (but not in general topological spaces), is equivalent to the topological notions of countable compactness and compactness defined via open covers.

Examples of compact metric spaces include the closed interval   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 if and only if 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  , there exists a "Lebesgue number"   such that every subset of   of diameter   is contained in some member of the cover.

Every compact metric space is second countable,[8] and is a continuous image of the Cantor set. (The latter result is due to Pavel Alexandrov and Urysohn.)

Locally compact and proper spacesEdit

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 if every closed ball   is compact. Proper spaces are locally compact, but the converse is not true in general.


A metric space   is connected if the only subsets that are both open and closed are the empty set and   itself.

A metric space   is path connected if for any two points   there exists a continuous map   with   and  . Every path connected space is connected, but the converse is not true in general.

There are also local versions of these definitions: locally connected spaces and locally path connected spaces.

Simply connected spaces are those that, in a certain sense, do not have "holes".

Separable spacesEdit

A metric space is separable space if it has a countable 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-countability and also to the Lindelöf property.

Pointed metric spacesEdit

If   is a metric space and   then   is called a pointed metric space, and   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   due to its similar behavior to zero in certain contexts.

Types of maps between metric spacesEdit

Suppose   and   are two metric spaces.

Continuous mapsEdit

The map   is continuous if it has one (and therefore all) of the following equivalent properties:

General topological continuity
for every open set   in  , the preimage   is open in  
This is the general definition of continuity in topology.
Sequential continuity
if   is a sequence in   that converges to  , then the sequence   converges to   in  .
This is sequential continuity, due to Eduard Heine.
ε-δ definition
for every   and every   there exists   such that for all   in   we have
This uses the (ε, δ)-definition of limit, and is due to Augustin Louis Cauchy.

Moreover,   is continuous if and only if it is continuous on every compact subset of  .

The 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 mapsEdit

The map   is uniformly continuous if for every   there exists   such that


Every uniformly continuous map   is continuous. The converse is true if   is compact (Heine–Cantor theorem).

Uniformly continuous maps turn Cauchy sequences in   into Cauchy sequences in  . For continuous maps this is generally wrong; for example, a continuous map from the open interval   onto the real line turns some Cauchy sequences into unbounded sequences.

Lipschitz-continuous maps and contractionsEdit

Given a real number  , the map   is K-Lipschitz continuous if


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

If  , then   is called a contraction. Suppose   and   is complete. If   is a contraction, then   admits a unique fixed point (Banach fixed-point theorem). If   is compact, the condition can be weakened a bit:   admits a unique fixed point if



The map   is an isometry if


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).


The map   is a quasi-isometry if there exist constants   and   such that


and a constant   such that every point in   has a distance at most   from some point in the image  .

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 equivalenceEdit

Given two metric spaces   and  :

  • 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 propertiesEdit

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   is the coarsest topology on   relative to which the metric   is a continuous map from the product of   with itself to the non-negative real numbers.

Distance between points and sets; Hausdorff distance and Gromov metricEdit

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   is a metric space,   is a subset of   and   is a point of  , we define the distance from   to   as

  where   represents the infimum.

Then   if and only if   belongs to the closure of  . Furthermore, we have the following generalization of the triangle inequality:


which in particular shows that the map   is continuous.

Given two subsets   and   of  , we define their Hausdorff distance to be

  where   represents the supremum.

In general, the Hausdorff distance   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   turns the set   of all non-empty compact subsets of   into a metric space. One can show that   is complete if   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.

Product metric spacesEdit

If   are metric spaces, and   is the Euclidean norm on  , then   is a metric space, where the product metric is defined by


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

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


An uncountable product of metric spaces need not be metrizable. For example,   is not first-countable and thus isn't metrizable.

Continuity of distanceEdit

In the case of a single space  , the distance map   (from the definition) is uniformly continuous with respect to any of the above product metrics  , and in particular is continuous with respect to the product topology of  .

Quotient metric spacesEdit

If M is a metric space with metric  , and   is an equivalence relation on  , then we can endow the quotient set   with a pseudometric. Given two equivalence classes   and  , we define


where the infimum is taken over all finite sequences   and   with  ,  ,  . In general this will only define a pseudometric, i.e.   does not necessarily imply that  . However, for some equivalence relations (e.g., those given by gluing together polyhedra along faces),   is a metric.

The quotient metric   is characterized by the following universal property. If   is a metric map between metric spaces (that is,   for all  ,  ) satisfying   whenever   then the induced function  , given by  , is a metric map  

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

Generalizations of metric spacesEdit

  • 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  , but otherwise satisfies the conditions of a metric, then it is called an extended metric and the corresponding space is called an  -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]

Metric spaces as enriched categoriesEdit

The ordered set   can be seen as a category by requesting exactly one morphism   if   and none otherwise. By using   as the tensor product and   as the identity, it becomes a monoidal category  . Every metric space   can now be viewed as a category   enriched over  :

  • Set  
  • For each   set  
  • The composition morphism   will be the unique morphism in   given from the triangle inequality  
  • The identity morphism   will be the unique morphism given from the fact that  .
  • Since   is a poset, all diagrams that are required for an enriched category commute automatically.

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

See alsoEdit


  1. ^ Rendic. Circ. Mat. Palermo 22 (1906) 1–74
  2. ^ B. Choudhary (1992). The Elements of Complex Analysis. New Age International. p. 20. ISBN 978-81-224-0399-2.
  3. ^ Huber, Klaus (January 1994) [1993-01-17, 1992-05-21]. "Codes over Gaussian Integers". IEEE Transactions on Information Theory. 40 (1): 207–216. doi:10.1109/18.272484. eISSN 1557-9654. ISSN 0018-9448. S2CID 195866926. IEEE Log ID 9215213. Archived (PDF) from the original on 2020-12-17. Retrieved 2020-12-17. [1][2] (1+10 pages) (NB. This work was partially presented at CDS-92 Conference, Kaliningrad, Russia, on 1992-09-07 and at the IEEE Symposium on Information Theory, San Antonio, TX, USA.)
  4. ^ Strang, Thomas; Dammann, Armin; Röckl, Matthias; Plass, Simon (October 2009). Using Gray codes as Location Identifiers (PDF). 6. GI/ITG KuVS Fachgespräch Ortsbezogene Anwendungen und Dienste (in English and German). Oberpfaffenhofen, Germany: Institute of Communications and Navigation, German Aerospace Center (DLR). CiteSeerX Archived (PDF) from the original on 2015-05-01. Retrieved 2020-12-16. Lay summary (PDF). (5/8 pages) [3]
  5. ^ Nathan Linial. Finite Metric Spaces—Combinatorics, Geometry and Algorithms, Proceedings of the ICM, Beijing 2002, vol. 3, pp573–586 Archived 2018-05-02 at the Wayback Machine
  6. ^ Open problems on embeddings of finite metric spaces, edited by Jirīı Matoušek, 2007 Archived 2010-12-26 at the Wayback Machine
  7. ^ Searcóid, p. 107.
  8. ^ "PlanetMath: a compact metric space is second countable". Archived from the original on 2009-02-05. Retrieved 2018-05-02.
  9. ^ Rudin, Mary Ellen. A new proof that metric spaces are paracompact Archived 2016-04-12 at the Wayback Machine. Proceedings of the American Mathematical Society, Vol. 20, No. 2. (Feb., 1969), p. 603.
  10. ^ "metric spaces are Hausdorff". PlanetMath.
  11. ^ Goreham, Anthony. Sequential convergence in Topological Spaces Archived 2011-06-04 at the Wayback Machine. Honours' Dissertation, Queen's College, Oxford (April, 2001), p. 14
  12. ^ a b Pascal Hitzler; Anthony Seda (2016-04-19). Mathematical Aspects of Logic Programming Semantics. CRC Press. ISBN 978-1-4398-2962-2.
  13. ^ "Partial metrics : welcome". Archived from the original on 2017-07-27. Retrieved 2018-05-02.

Further readingEdit

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.

External linksEdit