Problem : Suppose that $X$ is a finite set, and is Hausdorff as a topological space. Show that $X$ is discrete (as a topological space).
Thoughts: I'm not even quite sure what the question is asking. I know the definition of a discrete topology is that a set is open in $X$ if it is a subset of $\mathcal P(X)$, so then can't the discrete topology be applied to any set and so any $X$ is discrete? Any help appreciated.
Since $X$ is Hausdorff, we know that singletons are closed. Therefore, all finite subsets of $X$ are closed (finite unions of closed sets are closed). Since all subsets of $X$ are finite, it follows that every subset of $X$ is closed. Using this it is not hard to see that every subset of $X$ is open.