Countable compactness in powers of topological groups and Ramsey theoretic variations of compactness

Abstract

Countable compactness need not be preserved by finite products. Motivated by compactness notions for maps indexed by finite subsets of $\omega$, we introduce cascade countable compactness for arbitrary barriers. For $\mathcal{B}=[\omega {]}^2$, this is the previously studied notion of being doubly countably compact. We show that, in ZFC, if $G$ is a Hausdorff topological group and $1\le k<\omega$, then $k$-cascade countable compactness of $G$ implies that $G^k$ is countably compact. Cascade countable compactness for the Schreier barrier implies that $G^{\omega }$ is countably compact. In contrast, we construct a Tychonoff space that is $n$-cascade countably compact for every $n<\omega$ but has a non-countably compact square, and a Hausdorff Boolean group $H$ that is $\mathcal{B}$-countably compact for every barrier $\mathcal{B}$ but whose square is not countably compact. We also construct a Hausdorff Boolean group without nontrivial convergent sequences that is $\mathcal{B}$-cascade countably compact for every barrier $\mathcal{B}$. Finally, we obtain a subspace $X\subseteq \beta \omega$ such that $X^{\kappa }$ is $n$-cascade countably compact for every $\kappa <\mathfrak{h}$ and every $n<\omega$, whereas $\exp X$ is not pseudocompact.