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 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 $\omega$.