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

TL;DR

This paper explores the topological structure of the space of all topologies on a set, revealing its compact subsets and Borel complexity through lattice-theoretic and Stone space techniques.

Contribution

It establishes an equivalence between closures of sublattices of the topology space and their completions, advancing understanding of its topological and lattice-theoretic properties.

Findings

Identifies infinite compact subsets including Stone-echech and one-point compactifications.

Analyzes Borel complexity of $Top(X)$.

Provides lattice-theoretic characterizations of topological closures.

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.