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:

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.

History

This section needs expansion with: Reasons for generalizing the Euclidean metric, first non-Euclidean metrics studied, consequences for mathematics. You can help by adding to it. (August 2011)

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.

Definition

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 spaces

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 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 spaces

Complete spaces

Main article: Complete metric space

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 spaces

Diameter of a set.
Diameter of a set.

See also: bounded 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 spaces

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

Connectedness

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 spaces

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 spaces

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 spaces

Suppose and are two metric spaces.

Continuous maps

Main article: Continuous function (topology)

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 maps

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 contractions

Main article: Lipschitz continuity

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

.

Isometries

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).

Quasi-isometries

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 equivalence

Given two metric spaces and :

Topological properties

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 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 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 spaces

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 distance

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 spaces

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 spaces

Metric spaces as enriched categories

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 :

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

See also

References

  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 10.1.1.398.9164. 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". planetmath.org. 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". www.dcs.warwick.ac.uk. Archived from the original on 2017-07-27. Retrieved 2018-05-02.

Further reading

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.