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10 Prove that a topological space is compact it and only it every ultra-

filter in it is convergent.

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10 Prove that a topological space is compact it and only it every ultra-

filter in it is convergent.

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Mathematics
2 months
2021-07-27T14:51:04+00:00
2021-07-27T14:51:04+00:00 2 Answers
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## Answers ( )

A topological space X is compact if and only if every ultrafilter in X is convergent. α1 ∩ททท∩ U c αn = (Uα1 ∪ททท∪ Uαn )c = ∅. … Let X be a topological space. A family B of open subsets of X is a basis for the topology of X if every open subset of X is a union of elements of B, explicitly: (1) Every U ∈ B is an open set.

A topological space X is compact if and only if every ultrafilter in X is convergent. α1 ∩ททท∩ U c αn = (Uα1 ∪ททท∪ Uαn )c = ∅. … Let X be a topological space. A family B of open subsets of X is a basis for the topology of X if every open subset of X is a union of elements of B, explicitly: (1) Every U ∈ B is an open set.