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See Topological space. Totally bounded A metric space M is totally bounded if, for every r > 0, there exist a finite cover of M by open balls of radius r. A metric space is compact if and only if it is complete and totally bounded. Totally disconnected A space is totally disconnected if it has no connected subset with more than one point ...
(In other words, E replaces the "size" ε, and a subset is of size E if its Cartesian square is a subset of E.) [4] The definition can be extended still further, to any category of spaces with a notion of compactness and Cauchy completion: a space is totally bounded if and only if its (Cauchy) completion is compact.
Isomorphisms between metric spaces are called isometries. Every metric space is also a topological space. A topological space is called metrizable, if it underlies a metric space. All manifolds are metrizable. In a metric space, we can define bounded sets and Cauchy sequences. A metric space is called complete if all Cauchy sequences converge ...
The space M is called precompact or totally bounded if for every r > 0 there is a finite cover of M by open balls of radius r. Every totally bounded space is bounded. To see this, start with a finite cover by r-balls for some arbitrary r. Since the subset of M consisting of the centers of these balls is finite, it has finite diameter, say D.
In other words, the inclusion in the bidual () = is isomorphic to the inclusion of the space of countably additive μ-a.c. bounded measures inside the space of all finitely additive μ-a.c. bounded measures.
The space of Borel probability measures on a compact Hausdorff space is compact for the vague topology, by the Alaoglu theorem. A collection of probability measures on the Borel sets of Euclidean space is called tight if, for any positive epsilon, there exists a compact subset containing all but at most epsilon of the mass of each of the ...
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The word "bounded" makes no sense in a general topological space without a corresponding metric. Boundary is a distinct concept; for example, a circle (not to be confused with a disk ) in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.