On powers of countably pracompact groups

TL;DR

This paper constructs examples of topological groups with specific countably pracompact properties at certain powers, assuming the existence of multiple incomparable selective ultrafilters, addressing a question posed by Comfort in 1990.

Contribution

The paper provides the first constructions of topological groups with prescribed countably pracompact powers at certain cardinals, under set-theoretic assumptions.

Findings

Constructs groups with countably pracompact powers at  = \, \u03ba^+ for  = ,  , assuming selective ultrafilters.

Shows existence of groups where the -th power is countably pracompact but the next power is not.

Addresses a longstanding question about the behavior of countably pracompact powers in topological 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.