On the quasi-component of pseudocompact abelian groups

TL;DR

This paper investigates the relationship between the quasi-component of pseudocompact abelian groups and their completions, providing characterizations and examples of minimal groups with specific quasi-component properties.

Contribution

It characterizes pairs of compact connected abelian groups and subgroups related to the quasi-components of minimal pseudocompact abelian groups and their completions.

Findings

Characterization of pairs (C,A) related to quasi-components.

Existence of minimal pseudocompact groups with non-dense quasi-components.

Construction of abelian pseudocompact groups with specified quasi-component properties.

Abstract

In this paper, we describe the relationship between the quasi-component q(G) of a (perfectly) minimal pseudocompact abelian group G and the quasi-component q(\widetilde G) of its completion. Specifically, we characterize the pairs (C,A) of compact connected abelian groups C and subgroups A such that A \cong q(G) and C \cong q(\widetilde G). As a consequence, we show that for every positive integer n or n=\omega, there exist plenty of abelian pseudocompact perfectly minimal n-dimensional groups G such that the quasi-component of G is not dense in the quasi-component of the completion of G.