Discrete Subsets in Topological Groups and Countable Extremally Disconnected Groups

TL;DR

The paper investigates properties of countable topological groups, showing that certain conditions imply the existence of specific discrete sets and linking the structure of extremally disconnected groups to the existence of rapid ultrafilters, thus answering longstanding questions.

Contribution

It proves that countable topological groups with non-rapid neighborhood filters contain specific discrete sets and demonstrates that countable extremally disconnected groups cannot exist in ZFC, resolving open problems.

Findings

Countable topological groups with non-rapid filters contain discrete sets with one nonisolated point.

Existence of a countable extremally disconnected group implies the existence of a rapid ultrafilter.

Such groups cannot be constructed within ZFC.

Abstract

It is proved that any countable topological group in which the filter of neighborhoods of the identity element is not rapid contains a discrete set with precisely one nonisolated point. This gives a negative answer to Protasov's question on the existence in ZFC of a countable nondiscrete group in which all discrete subsets are closed. It is also proved that the existence of a countable nondiscrete extremally disconnected group implies the existence of a rapid ultrafilter and, hence, a countable nondiscrete extremally disconnected group cannot be constructed in ZFC.