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 2 P ( X ) . Via this identification then, Top ( X ) naturally inherits the subspace topology from 2 P ( X ) . Extending ideas of Frink (1942), we apply lattice-theoretic methods to establish an equivalence between the topological closures of sublattices of 2 P ( 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.