On the cardinality of Extremally Disconnected Groups with Linear Topology

TL;DR

This paper investigates the size and properties of extremally disconnected groups with linear topology, showing that certain large groups imply the existence of smaller ones with similar properties.

Contribution

It establishes a link between large extremally disconnected groups of Ulam nonmeasurable cardinality and smaller groups of size at most continuum with linear topology.

Findings

Existence of large extremally disconnected groups implies smaller ones of size at most continuum.

Linear topology's role in the structure of extremally disconnected groups.

Reduction of problem size from large to continuum cardinality.

Abstract

A group topology is said to be linear if open subgroups form a base of neighborhoods of the identity element. It is proved that the existence of a nondiscrete extremally disconnected group of Ulam nonmeasurable cardinality with linear topology implies that of a nondiscrete extremally disconnected group of cardinality at most $2^\omega$ with linear topology.