# Finite Hausdorff spaces

Is a finite Hausdorff space necessarily discrete?

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Just thought of another answer to this. Any topology on a finite set is compact. Any map from a discrete topology is continuous. Hence by the famous theorem on maps from compact spaces into Hausdorff spaces, the identity map on a finite space is a homeomorphism from the discrete topology to the given Hausdoff topology.

A certain phrase involving sledgehammers and walnuts springs to mind, though.

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LOL, that was funny (I mean about the walnuts...) –  Carlo Von Schnitzel May 11 '10 at 7:16