It remains true ( that is, the proof does not require choice ) that every precompact space is totally bounded; in other words, if the completion of a space is compact, then that space is totally bounded.
22.
It remains true ( that is, the proof does not require choice ) that every precompact space is totally bounded; in other words, if the completion of a space is compact, then that space is totally bounded.
23.
Then it becomes a theorem that a space is totally bounded if and only if it is precompact . ( Separating the definitions in this way is useful in the absence of the axiom of choice; see the next section .)
24.
But it is no longer true ( that is, the proof requires choice ) that every totally bounded space is precompact; in other words, the completion of a totally bounded space might not be compact in the absence of choice.
25.
But it is no longer true ( that is, the proof requires choice ) that every totally bounded space is precompact; in other words, the completion of a totally bounded space might not be compact in the absence of choice.
26.
And in the special case of an arbitrary compact metric space " X " every bounded subspace of an arbitrary metric space " Y " aimed at " X " is totally bounded ( i . e . its metric completion is compact ).
27.
X ^ \ star is defined as the space of all linear continuous functionals f : X \ to \ mathbb { C } endowed with the topology of uniform convergence on totally bounded sets in " X ", and the " second dual space"
28.
There is a complementary relationship between total boundedness and the process of Cauchy completion : A uniform space is totally bounded if and only if its Cauchy completion is totally bounded . ( This corresponds to the fact that, in Euclidean spaces, a set is bounded if and only if its closure is bounded .)
29.
There is a complementary relationship between total boundedness and the process of Cauchy completion : A uniform space is totally bounded if and only if its Cauchy completion is totally bounded . ( This corresponds to the fact that, in Euclidean spaces, a set is bounded if and only if its closure is bounded .)
30.
For any two stereotype spaces X and Y the " stereotype space of operators " \ text { Hom } ( X, Y ) from X into Y, is defined as the pseudosaturation of the space \ text { L } ( X, Y ) of all linear continuous maps \ varphi : X \ to Y endowed with the topology of uniform convergeance on totally bounded sets.
