A solution to Comfort's question on the countable compactness of powers of a topological group
Volume 186 / 2005
Fundamenta Mathematicae 186 (2005), 1-24
MSC: Primary 54B10, 54A35; Secondary 22A05, 03E50, 54G20.
DOI: 10.4064/fm186-1-1
Abstract
In 1990, Comfort asked Question $477$ in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) 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?
Hart and van Mill showed in 1991 that $\alpha=2$ answers this question affirmatively under ${\rm MA_{countable}}$. Recently, Tomita showed that every finite cardinal answers Comfort's question in the affirmative, also from ${\rm MA_{countable}}$. However, the question has remained open for infinite cardinals.
We show that the existence of $2^{\mathfrak c}$ selective ultrafilters $+$ $2^{\mathfrak c}=2^{<2^{\mathfrak c}}$ implies a positive answer to Comfort's question for every cardinal $\kappa \leq 2^{\mathfrak c}$. Thus, it is consistent that $\kappa$ can be a singular cardinal of countable cofinality. In addition, the groups obtained have no non-trivial convergent sequences.
