## Proof That a Finite Subset of a Hausdorff Space is Closed

Theorem

Ifis a Haudorff space and ifis a finite subset ofthenis closed.

Proof

Letand

A neighbourhoodofexists such that

Henceandis open andis closed.

By induction, any finite subsetis closed.

Theorem

Ifis a Haudorff space and ifis a finite subset ofthenis closed.

Proof

Letand

A neighbourhoodofexists such that

Henceandis open andis closed.

By induction, any finite subsetis closed.