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, $σ$-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 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.