Semitopological modules

TL;DR

This paper investigates semitopological modules over topological rings, exploring their completions, modifications, and the conditions under which free modules are topological, including counterexamples.

Contribution

It introduces the concept of semitopological modules over topological rings and analyzes their properties, including construction methods and conditions for topological module structures.

Findings

Construction of free semitopological modules over topological spaces

Identification of conditions when free modules are topological

Existence of Tychonoff spaces with non-topological free modules

Abstract

Given a topological ring $R$, we study semitopological $R$-modules, construct their completions, Bohr and borno modifications. For every topological space $X$, we construct the free (semi)topological $R$-module over $X$ and prove that for a $k$-space $X$ its free semitopological $R$-module is a topological $R$-module. Also we construct a Tychonoff space $X$ whose free semitopological $R$-module is not a topological $R$-module.