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Show that the intersection of a family of topologies of X is a topology for X.

Show that the intersection of a family of topologies of X is a topology for X.

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Answer #1

If {Tα} is a family of topologies on X, then Tα is a topology on X

. Proof. (i)Since ∅ and X are in each Tα, they must also be elements of Tα.

(ii) if C ⊂ Tα, then C ⊂ Tα for all α; it follows that C ∈ Tα for all α, and hence C ∈ Tα.

(iii)Finally, if C is a finite subcollection ofTα, then C ∈ Tα for all α, and therefore C ∈ Tα

Hence intersection is topology on X

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