A solution to Comfort's question on the countable compactness of powers of a topological group

Artur Hideyuki Tomita¹

¹ Departamento de Matemática Instituto de Matemática e Estatística Universidade de São Paulo Caixa Postal 66281 CEP 05315-970, São Paulo, Brazil

Fundamenta Mathematicae, Tome 186 (2005) no. 1, pp. 1-24.

Résumé

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.

Détail

DOI : 10.4064/fm186-1-1

Keywords: comfort asked question survey book problems topology there every necessarily infinite cardinal number alpha leq mathfrak topological group gamma countably compact cardinals gamma alpha alpha countably compact hart van mill showed alpha answers question affirmatively under countable recently tomita showed every finite cardinal answers comforts question affirmative countable however question has remained infinite cardinals existence mathfrak selective ultrafilters mathfrak mathfrak implies positive answer comforts question every cardinal kappa leq mathfrak consistent kappa singular cardinal countable cofinality addition groups obtained have non trivial convergent sequences