A metric space $X$ that, for any $\epsilon>0$, can be represented as the union of a finite number of sets with diameters smaller than $\epsilon$. An equivalent condition is the following: For each $\epsilon>0$ there exists in $X$ a finite $\epsilon$-net, i.e. a finite set $A$ such that the distance of each point of $X$ from some point of $A$ is less than $\epsilon$. Totally-bounded spaces are those, and only those, metric spaces that can be represented as subspaces of compact metric spaces (cf. Compact space). The metric totally-bounded spaces, considered as topological spaces, exhaust all regular spaces (cf. Regular space) with a countable base. A subspace of a Euclidean space is totally bounded if and only if it is bounded. The converse is not true: An infinite space in which the distance between any two points is one, as well as a sphere and a ball of a Hilbert space, are bounded, but not totally bounded, metric spaces. The significance of the concept of a totally-bounded space may be illustrated by the following theorem: A metric space is a compactum if and only if it is totally bounded and complete. The metric completion of a metric totally-bounded space is compact. The image of a totally-bounded space under a uniformly continuous mapping is a totally-bounded space.
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Totally-bounded space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Totally-bounded_space&oldid=31018