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

100%

Tomita A.

Fundamenta Mathematicae

2005

tom 186

nr 1

1-24

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.

A group topology on the free abelian group of cardinality 𝔠 that makes its square countably compact

63%

Boero A., Tomita A.

Fundamenta Mathematicae

2011

tom 212

nr 3

235-260

Under 𝔭 = 𝔠, we prove that it is possible to endow the free abelian group of cardinality 𝔠 with a group topology that makes its square countably compact. This answers a question posed by Madariaga-Garcia and Tomita and by Tkachenko. We also prove that there exists a Wallace semigroup (i.e., a countably compact both-sided cancellative topological semigroup which is not a topological group) whose square is countably compact. This answers a question posed by Grant.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Tomita A.

1 Boero A.

1 2011

1 2005

Wyniki wyszukiwania

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

A group topology on the free abelian group of cardinality 𝔠 that makes its square countably compact