Combining these theorems, a uniform space is totally bounded if and only if its completion is compact.
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A set is totally bounded if, given any positive radius, it is covered by finitely many balls of that radius.
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A compact set in fact can be covered by finitely many balls of arbitrary small radius, because it is totally bounded.
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*PM : totally bounded uniform space, id = 8958 new !-- WP guess : totally bounded uniform space-- Status:
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*PM : totally bounded uniform space, id = 8958 new !-- WP guess : totally bounded uniform space-- Status:
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:* X is quasibarreled \ Longleftrightarrow in X ^ \ star if a set T is absorbed by each barrel, then T is totally bounded;
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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 completion is compact.
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Alternatively, " X " is " right "-totally bounded if and only if it satisfies the definition for topological abelian groups above, using " right " translates.
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X ^ \ star is defined as the space of continuous linear functionls X'endowed with the topology of uniform convergence on totally bounded sets in X ( and the " stereotype second dual space"
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A topological group " X " is " left "-totally bounded if and only if it satisfies the definition for topological abelian groups above, using " left " translates.
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