Question

10 Prove that a topological space is compact it and only it every ultra-
filter in it is convergent.​

Answer 1

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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.

Answer 2

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.