Locally precompact groups: (Local) realcompactness and connectedness

TL;DR

This paper classifies locally precompact groups based on various topological properties like realcompactness and connectedness, and characterizes when abelian groups appear as quasi-components in larger groups.

Contribution

It provides a comprehensive classification of locally precompact groups with specific topological features and characterizes abelian groups as quasi-components.

Findings

Classified locally precompact groups by topological properties

Proved conditions for abelian groups to be quasi-components

Connectedness and realcompactness criteria established

Abstract

A theorem of A. Weil asserts that a topological group embeds as a (dense) subgroup of a locally compact group if and only if it contains a non-empty precompact open set; such groups are called locally precompact. Within the class of locally precompact groups, the authors classify those groups with the following topological properties: Dieudonn\'e completeness; local realcompactness; realcompactness; hereditary realcompactness; connectedness; local connectedness; zero-dimensionality. They also prove that an abelian locally precompact group occurs as the quasi-component of a topological group if and only if it is precompactly generated, that is, it is generated algebraically by a precompact subset.