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

Artur Hideyuki Tomita

Fundamenta Mathematicae, Tome 185 (2005), p. 1-24 / Harvested from The Polish Digital Mathematics Library

Résumé

In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number $α \leq 2^{}$ , 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 $M A_{c o u n t a b l e}$ . Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from $M A_{c o u n t a b l e}$ . However, the question has remained open for infinite cardinals. We show that the existence of $2^{}$ selective ultrafilters + $2^{} = 2^{< 2^{}}$ implies a positive answer to Comfort’s question for every cardinal $κ \leq 2^{}$ . 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.

Publié le : 2005-01-01

Zbl 1084.54003

EUDML-ID : urn:eudml:doc:282747

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     title = {A solution to Comfort's question on the countable compactness of powers of a topological group},
     journal = {Fundamenta Mathematicae},
     volume = {185},
     year = {2005},
     pages = {1-24},
     zbl = {1084.54003},
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Artur Hideyuki Tomita. A solution to Comfort's question on the countable compactness of powers of a topological group. Fundamenta Mathematicae, Tome 185 (2005) pp. 1-24. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm186-1-1/