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Informally, a metric space is complete if it has no "missing points": every sequence that looks like it should converge to something actually converges. To make this precise: a sequence (x n) in a metric space M is Cauchy if for every ε > 0 there is an integer N such that for all m, n > N, d(x m, x n) < ε.
The space Aim(X) is injective (hyperconvex in the sense of Aronszajn-Panitchpakdi) – given a metric space M, which contains Aim(X) as a metric subspace, there is a canonical (and explicit) metric retraction of M onto Aim(X) (Holsztyński 1966).
Thus metric spaces together with metric maps form a category Met. Met is a subcategory of the category of metric spaces and Lipschitz functions. A map between metric spaces is an isometry if and only if it is a bijective metric map whose inverse is also a metric map. Thus the isomorphisms in Met are precisely the isometries.
The function d is a metric on M, and d(x, y) is the distance between x and y. The collection of all open balls of M is a base for a topology on M; this is the topology on M induced by d. Every metric space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metric space is first-countable. Metrizable/Metrisable
The product of a finite set of metric spaces in Met is a metric space that has the cartesian product of the spaces as its points; the distance in the product space is given by the supremum of the distances in the base spaces. That is, it is the product metric with the sup norm. However, the product of an infinite set of metric spaces may not ...
For any metric space M, it is possible to construct a complete metric space M′ (which is also denoted as ¯), which contains M as a dense subspace. It has the following universal property : if N is any complete metric space and f is any uniformly continuous function from M to N , then there exists a unique uniformly continuous function f ...
Hyperbolic metric space (3 P) M. Metric linear spaces (1 C) R. Riemannian manifolds (41 P) Pages in category "Metric spaces"
The space (0,1) ⊂ R, the open unit interval, is not a complete metric space with its usual metric inherited from R, but it is completely metrizable since it is homeomorphic to R. [6] The space Q of rational numbers with the subspace topology inherited from R is metrizable but not completely metrizable. [7]