When is the Sum of Two Closed Subgroups Closed in a Locally Compact Abelian Group

TL;DR

This paper classifies locally compact abelian groups where the sum of any two closed subgroups is closed, extending Mukhin's 1970 results to a complete classification of totally disconnected groups.

Contribution

It provides a complete classification of totally disconnected locally compact abelian groups with closed sums of any two closed subgroups, extending previous work by Mukhin.

Findings

Characterization of totally disconnected groups with closed subgroup sums

Extension of Mukhin's 1970 results

Complete classification of such groups

Abstract

Locally compact abelian groups are classified in which the sum of any two closed subgroups is itself closed. This amounts to reproving and extending results by Yu.~N.~Mukhin from 1970. Namely we contribute a complete classification of all totally disconnected \lca\ groups with $X+Y$ closed for any closed subgroups $X$ and $Y$.