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

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.

23 pages