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

Abstract

A topological group is locally pseudocompact if it contains a non-empty open set with pseudocompact closure. In this paper, we prove that if G G is a group with the property that every closed subgroup of G G is locally pseudocompact, then G 0 G_0 is dense in the component of the completion of G G , and G / G 0 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 G/G_0 may fail to be zero-dimensional even for totally minimal pseudocompact groups.