Topological groups with strong disconnectedness properties

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 Lindelöf basically disconnected topological group which is not a P-space is equivalent to the existence of a Boolean basically disconnected Lindelöf 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 ω.