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[0, 1] 2 is a totally bounded space because for every ε > 0, the unit square can be covered by finitely many open discs of radius ε. A metric space (,) is totally bounded if and only if for every real number >, there exists a finite collection of open balls of radius whose centers lie in M and whose union contains M.
The metric space (M, d) is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself. Total boundedness implies boundedness. For subsets of R n the two are equivalent. A metric space is compact if and only if it is complete and totally bounded. A subset of Euclidean space R n is compact if and only if it is closed and
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.
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.
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A function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M such that | | for all x in X. A function that is not bounded is said to be unbounded. Sometimes, if f(x) ≤ A for all x in X, then the function is said to be bounded above by A. On ...
A space where is controlled is called a bounded space. Such a space is coarsely equivalent to a point. A metric space with the bounded coarse structure is bounded (as a coarse space) if and only if it is bounded (as a metric space).