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It can be proved that if a topological space with cofinite topology is Hausdorff then it's finite. For instance see drhab's post here. I was wondering whether the converse is also true? That's if $X$ is finite then $\{x\}, \{y\}$ are open sets and $\{x\}\cap \{y\}=\emptyset$ for any $x,y \in X, x\ne y$. Thus $X$ is Hausdorff. Is this correct?

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Yes, that's correct - the key point of course being that $\{x\}$ and $\{y\}$ are open since the whole space is finite, so the complement of any set is finite; that is, the cofinite topology on a finite set is discrete.

Noah Schweber
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