On zero-dimensionality and the connected component of locally pseudocompact groups

TL;DR

This paper investigates the structure of locally pseudocompact groups, showing that under certain conditions, their connected components are dense and their quotients are zero-dimensional, with examples illustrating exceptions.

Contribution

It proves that if all closed subgroups are locally pseudocompact, then the group's connected component is dense and the quotient is zero-dimensional, and provides counterexamples.

Findings

G_0 is dense in the component of the completion of G

G/G_0 is zero-dimensional under certain conditions

Counterexamples show G/G_0 may not be zero-dimensional in general

Abstract

A topological group is locally pseudocompact if it contains a non-empty open set with pseudocompact closure. In this note, we prove that if G is a group with the property that every closed subgroup of G is locally pseudocompact, then G_0 is dense in the component of the completion of G, and G/G_0 is zero-dimensional. We also provide examples of hereditarily disconnected pseudocompact groups with strong minimality properties of arbitrarily large dimension, and thus show that G/G_0 may fail to be zero-dimensional even for totally minimal pseudocompact groups.