Topologies as points within a Stone space: Lattice theory meets topology

For a non-empty set X, the collection Top(X) of all topologies on X sits inside the Boolean lattice P(P(X)) (when ordered by set-theoretic inclusion) which in turn can be naturally identified with the Stone space 2P(X). Via this identification then, Top(X) naturally inherits the subspace topology from 2P(X). Extending ideas of Frink (1942), we apply lattice-theoretic methods to establish an equivalence between the topological closures of sublattices of 2P(X) and their (completely distributive) completions. We exploit this equivalence when searching for countably infinite compact subsets within Top(X) and in crystallizing the Borel complexity of Top(X). We exhibit infinite compact subsets of Top(X) including, in particular, copies of the Stone–Čech and one-point compactifications of discrete spaces.