Menger remainders of topological groups

TL;DR

This paper investigates how combinatorial covering properties like Menger, Scheepers, and Hurewicz influence the structure of remainders in topological groups, revealing conditions under which these remainders exhibit specific compactness properties.

Contribution

It establishes new characterizations of remainders with these properties, including a characterization of Hurewicz remainders as σ-compact and the independence results related to Scheepers remainders.

Findings

Hurewicz remainders are σ-compact

Existence of Scheepers non-σ-compact remainders depends on CH

Aims to prove a dichotomy for Menger remainders

Abstract

In this paper we discuss what kind of constrains combinatorial covering properties of Menger, Scheepers, and Hurewicz impose on remainders of topological groups. For instance, we show that such a remainder is Hurewicz if and only it is $\sigma$-compact. Also, the existence of a Scheepers non-$\sigma$-compact remainder of a topological group follows from CH and yields a $P$-point, and hence is independent of ZFC. We also make an attempt to prove a dichotomy for the Menger property of remainders of topological groups in the style of Arhangel'skii.