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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:
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.

In any metric space M we can define the open balls as the sets of the form
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 realvalued 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 realvalued Lipschitzcontinuous map defined on a subset of a metric space can be extended to a Lipschitzcontinuous map on the whole space.
Two metric spaces (M_{1}, d_{1}) and (M_{2}, d_{2}) are called isometrically isomorphic iff there exists a bijective function f : M_{1} → M_{2} with the property d_{2}(f(x), f(y)) = d_{1}(x, y) for all x, y in M_{1}. 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.
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
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:
If one drops property 3, one obtains pseudometric spaces; if one drops property 4 instead, one obtains quasimetric spaces[?].
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