Topologies on X as points within 2P(X)

A topology on a nonempty set X specifies a natural subset of P(X). By identifying P(P(X)) with the totally disconnected compact Hausdorff space 2P(X), the lattice Top(X) of all topologies on X is a natural subspace therein. We investigate topological properties of Top(X) and give sufficient model-theoretic conditions for a general subspace of 2P(X) to be compact.