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

• Autorzy: Artur Hideyuki Tomita
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 03E50: Continuum hypothesis and Martin's axiom
• 54B10: Product spaces
• 54G20: Counterexamples
• 54A35: Consistency and independence results
• 22A05: Structure of general topological groups

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2005
• Tom: 186
• Numer: 1
• Strony: 1-24

Daty

wydano

2005

Twórcy

autor

Artur Hideyuki Tomita

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

Identyfikatory

DOI

10.4064/fm186-1-1