Totally disconnected semigroup compactifications: non-introversion of the full Boolean algebra of clopen sets

Abstract

In terms of the existence of a single clopen set and two related nets, we characterize when the full Boolean algebra \({\mathfrak {B}}(G)\) of clopen subsets of a topological group G is left introverted. We employ this characterization to show that when G is a first countable, \(\sigma \)-compact, totally disconnected locally compact group, \({\mathfrak {B}}(G)\) is left introverted if and only if G is compact or discrete, thus providing a strong positive answer to a question posed in Stephens and Stokke (Q J Math 74:301–326 2023). Examples of clopen sets and nets witnessing our non-introversion theorem are presented. Some hereditary properties of left introversion of \({\mathfrak {B}}(G)\) are proved and then employed to extend our main result to other classes of topological groups.