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2005 | 186 | 1 | 1-24
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A solution to Comfort's question on the countable compactness of powers of a topological group

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In 1990, Comfort asked Question 477 in the survey book "Open Problems in Topology": Is there, for every (not necessarily infinite) cardinal number $α ≤ 2^{𝔠}$, a topological group G such that $G^γ$ is countably compact for all cardinals γ < α, but $G^α$ is not countably compact?
Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under $MA_{countable}$. Recently, Tomita showed that every finite cardinal answers Comfort's question in the affirmative, also from $MA_{countable}$. However, the question has remained open for infinite cardinals.
We show that the existence of $2^{𝔠}$ selective ultrafilters + $2^{𝔠} = 2^{<2^{𝔠}}$ implies a positive answer to Comfort's question for every cardinal $κ ≤ 2^{𝔠}$. Thus, it is consistent that κ can be a singular cardinal of countable cofinality. In addition, the groups obtained have no non-trivial convergent sequences.
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  • Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, CEP 05315-970, São Paulo, Brazil
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm186-1-1
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