As I understand the definition of a subspace, I can simply take any subset of a topological space and promote it to a subspace by endowing it with the subspace topology. Then in this case every subset of my Hausdorff space, endowed with the subspace topology for whatever topology the space has, is compact. Therefore every subspace is a closed subset. I would like for this generalization to be true since from here I think I can go on to talk about the topology on my Hausdorff space being the discrete topology and the space being finite.
Let $A \subset X$. Then by assumption, $A$ (in the subspace topology, what else?) is compact. A compact subspace in a Hausdorff space is closed. So $A$ is closed. So all subsets of $X$ are closed, so all subsets have a closed complement as well, so all subsets are open too. Hence $X$ is discrete etc.