On proper compactifications of topological groups

TL;DR

This paper explores graph and Ellis methods for constructing proper compactifications of topological groups, analyzing their algebraic and topological properties, with applications to permutation and automorphism groups.

Contribution

It provides detailed descriptions of various compactifications of topological groups and demonstrates their use in studying topological properties of remainders, using dichotomy theorems.

Findings

Descriptions of Roelcke, Ellis, WAP, and graph compactifications.

Effective use of compactification descriptions in topological property analysis.

Examples involving permutation groups and automorphism groups.

Abstract

In the present paper, we examine in detail the method of "graph compactifications" of topological groups. The graph and Ellis methods of constructing proper compactifications of topological groups are applied for the investigation of possible extensions of algebraic operations on a topological group to its compactifications, and give descriptions of Roelcke, Ellis, WAP, and graph compactifications of topological groups. Additionally, using dichotomy theorems of A.V.Arhangelskii, we show that the description of compactifications can be effectively used in the investigation of topological properties of their remainders. As examples, subgroups of the permutation group (in the permutation topology) and the automorphism group of a LOTS (in the topology of pointwise convergence) are examined.