$Top(X)$ within $\px$ ]{When lattices meet topology: $Top(X)$ within $\px$.}

Abstract

For a non-empty set $X$, the collection $Top(X)$ of all topologies on $X$ sits inside the Boolean lattice $\PP(\PP(X))$ (when ordered by set-theoretic inclusion) which in turn can be naturally identified with the Stone space $\px$. Via this identification then, $Top(X)$ naturally inherits the subspace topology from $\px$ (see \cite{TopX1}). Extending ideas of Frink \cite{MR0006496}, we establish an equivalence between the topological closures of sublattices of $\px$ and their (completely distributive) completions. We exploit this equivalence when searching for countably infinite compact subsets within $Top(X)$ and in crystalizing the Borel complexity of $Top(X)$. We exhibit infinite compact subsets of $Top(X)$ including, in particular, copies of the Stone-\v{C}ech and one-point compactifications of discrete spaces.