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:

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

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.

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.

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

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

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.

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.

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.)

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.

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".

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.

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.

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

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).

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.

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.

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.

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

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

This is reprinted (with author commentary) at Also (with an author commentary) in Enriched categories in the logic of geometry and analysis. Repr. Theory Appl. Categ. No. 1 (2002), 1–37.