On powers of countably pracompact groups

Abstract

In 1990, Comfort asked: is there, for every cardinal number $α\leq 2^{\mathfrak{c}}$, a topological group $G$ such that $G^γ$ is countably compact for all cardinals $γ<α$, but $G^α$ is not countably compact? A similar question can also be asked for countably pracompact groups: for which cardinals $α$ is there a topological group $G$ such that $G^γ$ is countably pracompact for all cardinals $γ< α$, but $G^α$ is not countably pracompact? In this paper we construct such group in the case $α= ω$, assuming the existence of $\mathfrak{c}$ incomparable selective ultrafilters, and in the case $α= κ^{+}$, with $ω\leq κ\leq 2^{\mathfrak{c}}$, assuming the existence of $2^{\mathfrak{c}}$ incomparable selective ultrafilters. In particular, under the second assumption, there exists a topological group $G$ so that $G^{2^\mathfrak{c}}$ is countably pracompact, but $G^{(2^{\mathfrak{c}})^{+}}$ is not countably pracompact, unlike the countably compact case.