Topologies on X as points within

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 2 P ( 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 2 P ( X ) to be compact.