On powers of countably pracompact groups

Abstract

In 1990, Comfort asked: is there, for every cardinal number $\alpha \leq 2^{\mathfrak{c}}$, a topological group $G$ such that $G^\gamma$ is countably compact for all cardinals $\gamma<\alpha$, but $G^\alpha$ is not countably compact? A similar question can also be asked for countably pracompact groups: for which cardinals $\alpha$ is there a topological group $G$ such that $G^{\gamma}$ is countably pracompact for all cardinals $\gamma < \alpha$, but $G^{\alpha}$ is not countably pracompact? In this paper we construct such group in the case $\alpha = \omega$, assuming the existence of $\mathfrak{c}$ incomparable selective ultrafilters, and in the case $\alpha = \kappa^{+}$, with $\omega \leq \kappa \leq 2^{\mathfrak{c}}$, 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.