"Topological Groups with Strong Disconnectedness Properties"
Sipacheva, OlgaThere is a whole hierarchy of classical strong disconnectedness properties: a space $X$ is maximal if it has no isolated points and any two disjoint subsets of $X$ have disjoint closures; $X$ is extremally disconnected if any two disjoint open subsets of $X$ have disjoint closures (or, equivalently, the closure of any open set in $X$ is open); $X$ is basically disconnected if the closure of any cozero set in $X$ is open; $X$ an $F$-space if any two disjoint cozero sets are completely (=functionally) separated in $X$; and, finally, $X$ is an $F'$-space if any two disjoint cozero sets in $X$ have disjoint closures. It is well known that all strong disconnectednesses badly affect homogeneity properties. Thus, it is natural to ask whether any of them can coexist with the property of being a topological group, which can be regarded as ultimate homogeneity. It is known that both the existence and nonexistence of nondiscrete maximal, as well as countable extremally disconnected, groups is consistent with ZFC. The existence problem for uncountable nondiscrete extremally disconnected, as well as for basically disconnected, $F$-, and $F'$-groups not being $P$-spaces, remains unsolved. The talk is devoted to tological groups whose underlying spaces are basically disconnected, $F$-, or $F'$-spaces but not $P$-spaces. It is proved, in particular, that the existence of an Abelian basically disconnected group which is not a $P$-space is equivalent to the existence of a nondiscrete Boolean basically disconnected group of countable pseudocharacter. It is also proved that free and free Abelian topological groups of zero-dimensional non-$P$-spaces are never $F'$-spaces and that the existence of free Boolean $F'$-groups is equivalent to that of selective ultrafilters on $\omega$.