Topological Groups with Strong Disconnectedness Properties

TL;DR

This paper investigates topological groups with specific disconnectedness properties, establishing equivalences between their existence and certain set-theoretic conditions, and analyzing their structural characteristics.

Contribution

It provides new equivalences linking the existence of certain topological groups with set-theoretic ultrafilters and characterizes their properties in terms of disconnectedness.

Findings

Existence of Lindelöf basically disconnected groups not being P-spaces is equivalent to certain ultrafilter conditions.

Free and free Abelian groups of zero-dimensional non-P-spaces are never F'-spaces.

Existence of a free Boolean F'-group not being a P-space is equivalent to the existence of selective ultrafilters.

Abstract

Topological groups whose underlying spaces are basically disconnected, $F$-, or $F'$-spaces but not $P$-spaces are considered. It is proved, in particular, that the existence of a Lindel\"of basically disconnected topological group which is not a $P$-space is equivalent to the existence of a Boolean basically disconnected Lindel\"of group of countable pseudocharacter, that free and free Abelian topological groups of zero-dimensional non-$P$-spaces are never $F'$-spaces, and that the existence of a free Boolean $F'$-group which is not a $P$-space is equivalent to that of selective ultrafilters on $\omega$.