Comfort's question on powers in $\mathbb Q ^{(2^\mathfrak c)}$ and a Wallace semigroup whose cube is countably compact

TL;DR

This paper explores the existence of special ultrafilters and their implications for constructing Wallace semigroups and torsion-free topological groups with countably compact powers, addressing Comfort's question.

Contribution

It establishes new connections between ultrafilter assumptions and the construction of specific topological algebraic structures with compactness properties.

Findings

Existence of Wallace semigroup with countably compact cube under ultrafilter assumptions

Construction of torsion-free topological groups with countably compact powers

Implications for Comfort's question on powers of topological groups

Abstract

We prove that the existence of $\mathfrak{c}$ incomparable selective ultrafilters implies the existence of a Wallace semigroup whose cube is countably compact. In addition, assuming the existence of $2^{\mathfrak c}$ incomparable selective ultrafilters and $2^{< 2^{\mathfrak{c}}} = 2^{\mathfrak{c}}$, we obtain torsion-free topological groups with respect to Comfort's question on the countable compactness of (infinite) powers of a topological group.