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Metric space

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A metric space is a space where a distance between points is defined. Formally, a metric space is a set of points M with an associated distance function (also called a metric) d : M M -> R (where R is the set of real numbers) that satisfies the conditions:

  1. For all x, y in M, d(x, y) ≥ 0
  2. For all x in M, d(x, x) = 0
  3. For all x, y in M, if d(x, y) = 0 then x = y.
  4. For all x, y in M, d(x, y) = d(y, x).
  5. For all x, y, z in M, d(x, z) ≤ d(x, y) + d(y, z); this is the triangle inequality.

These axioms express intuitive notions about the concept of "distance": distances between different spots are positive and the distance between x and y is the same as the distance between y and x. The triangle inequality means that if you go from x to z directly, that is no longer than going first from x to y, and then from y to z. In Euclidean geometry, this is easy to see. Metric spaces allow this concept to be extended to a more abstract setting.

A metric space in which every Cauchy sequence has a limit is said to be complete.

Table of contents

Examples

  • The real numbers with the distance function d(x, y) = |y - x| given by the absolute value, and more generally Euclidean n-space with the Euclidean distance, are complete metric spaces.
  • More generally still, any normed vector space is a metric space by defining d(x, y) = ||y - x||. If such a space is complete, we call it a Banach space.
  • If X is some set and M is a metric space, then the set of all bounded functions f : X -> M (i.e. those functions whose image is a bounded subset of M) can be turned into a metric space by defining d(f, g) = supx in X d(f(x), g(x)) for any bounded functions f and g. If M is complete, then this space is complete as well.
  • If X is a topological (or metric) space and M is a metric space, then the set of all bounded continuous functions from X to M forms a metric space if we define the metric as above: d(f, g) = supx in X d(f(x), g(x)) for any bounded continuous functions f and g. If M is complete, then this space is complete as well.
  • If M is a connected Riemannian manifold, then we can turn M into a metric space by defining the distance of two points as the infimum of the lengths of the paths (continuously differentible curves) connecting them.
  • If G is an undirected connected graph, then the set V of vertices of G can be turned into a metric space by defining d(x, y) to be the length of the shortest path connecting the vertices x and y.
  • If M is a metric space, we can turn the set K(M) of all compact subsets of M into a metric space by defining the Hausdorff distance d(X, Y) = inf{r : for every x in X there exists a y in Y with d(x, y) < r and for every y in Y there exists an x in X such that d(x, y) < r)}. In this metric, two elements are close to each other if every element of one set is close to some element of the other set. One can show that K(M) is complete if M is complete.

Further defintions and properties

In any metric space M we can define the open balls as the sets of the form

B(x; r) = {y in M : d(x,y) < r},
where x is in M and r is a positive real number, called the radius of the ball. A subset of M which is a union of (finitely or infinitely many) open balls is called an open set. The complement of an open set is called closed. Every metric space is automatically a topological space, the topology being the set of all open sets. A topological space which can arise in this way from a metric space is called a metrizable space; see the article on metrization theorems for further details.

A metric space M is called bounded if there exists some number r > 0 such that d(x,y) ≤ r for all x and y in M (not to be confused with "finite", which refers to the number of elements, not to how far the set extends; finiteness implies boundedness, but not conversely). The space M is called totally bounded if for every r > 0 there exist finitely many open balls of radius r whose union equals M. It is not difficult to see that every totally bounded space is bounded. It can be shown that a metric space is compact if and only if it is complete and totally bounded.

By restricting the metric, any subset of a metric space is a metric space itself. We call such a subset complete, bounded, totally bounded or compact if it, considered as a metric space, has the corresponding property.

Metric spaces are paracompact Hausdorff spaces 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.

Two metric spaces (M1, d1) and (M2, d2) are called isometrically isomorphic iff there exists a bijective function f : M1M2 with the property d2(f(x), f(y)) = d1(x, y) for all x, y in M1. In this case, the two spaces are essentially identical. An isometry is a function f with the stated property, which is then necessarily injective but may fail to be surjective.

Every metric space is isometrically isomorphic to a closed subset of some normed vector space. Every complete metric space is isometrically isomorphic to a closed subset of some Banach space.

Distance between points and sets

If (M,d) is a metric space, S is a subset of M and x is a point of M, we define the distance from x to S as

d(x,S) = inf {d(x,s) : sS}
Then d(x, S) = 0 if and only if x belongs to the closure of S. Furthermore, we have the following generalization of the triangle inequality:
d(x,S) ≤ d(x,y) + d(y,S)
which in particular shows that the map x |-> d(x,S) is continuous.

Alternative axiom systems

The property 1 (d(x, y) ≥ 0) follows from properties 4 and 5 and does not have to be required separately.

Some authors use the extended real number line and allow the distance function d to attain the value ∞. Every such metric can be rescaled to a finite metric (using d'(x, y) = d(x, y) / (1 + d(x, y)) or d''(x, y) = min(1, d(x, y))) and the two concepts of metric space are therefore equivalent.

Some metrics satisfy a stronger version of the triangle inequality:

  • For all x, y, z in M, d(x, z) ≤ max(d(x, y), d(y, z))
These metrics are called super-metrics. An equivalent condition is that every triangle has at least two equal sides. The p-adic numbers are a complete super-metric space.

If one drops property 3, one obtains pseudometric spaces; if one drops property 4 instead, one obtains quasimetric spaces[?].


See also: contraction mapping.



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