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

摘要

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-Cech and one-point compactifications of discrete spaces.