Nonexistence of Countable Extremally Disconnected Groups with Many Open Subgroups

TL;DR

This paper proves that certain countable extremally disconnected Boolean topological groups with many open subgroups imply the existence of a rapid ultrafilter, linking topological group properties to ultrafilter existence.

Contribution

It establishes a connection between the structure of countable extremally disconnected groups and the existence of rapid ultrafilters, a novel result in topological group theory.

Findings

Existence of such groups implies rapid ultrafilters.

No such groups exist without the assumption of rapid ultrafilters.

Links between topological group properties and ultrafilter theory.

Abstract

It is proved that the existence of a countable extremally disconnected Boolean topological group containing a family of open subgroups whose intersection has empty interior implies the existence of a rapid ultrafilter.