2012/02/28 by Jorge Bruno, Bruno, Jorge L., Aisling McCluskey +1
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.1202.6180
openalex publication_date 2012/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \citeTopX1). Extending ideas of Frink \citeMR0006496, 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-Čech and one-point compactifications of discrete spaces.