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

Artur Hideyuki Tomita — 2005

Fundamenta Mathematicae

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 , 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...

On finite powers of countably compact groups

Artur Hideyuki Tomita — 1996

Commentationes Mathematicae Universitatis Carolinae

We will show that under M A c o u n t a b l e for each k there exists a group whose k -th power is countably compact but whose 2 k -th power is not countably compact. In particular, for each k there exists l [ k , 2 k ) and a group whose l -th power is countably compact but the l + 1 -st power is not countably compact.

The existence of initially ω 1 -compact group topologies on free Abelian groups is independent of ZFC

Artur Hideyuki Tomita — 1998

Commentationes Mathematicae Universitatis Carolinae

It was known that free Abelian groups do not admit a Hausdorff compact group topology. Tkachenko showed in 1990 that, under CH, a free Abelian group of size admits a Hausdorff countably compact group topology. We show that no Hausdorff group topology on a free Abelian group makes its ω -th power countably compact. In particular, a free Abelian group does not admit a Hausdorff p -compact nor a sequentially compact group topology. Under CH, we show that a free Abelian group does not admit a Hausdorff...

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

Ana Carolina BoeroArtur Hideyuki Tomita — 2011

Fundamenta Mathematicae

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.

Countable compactness and p -limits

Salvador García-FerreiraArtur Hideyuki Tomita — 2001

Commentationes Mathematicae Universitatis Carolinae

For M ω * , we say that X is quasi M -compact, if for every f : ω X there is p M such that f ¯ ( p ) X , where f ¯ is the Stone-Čech extension of f . In this context, a space X is countably compact iff X is quasi ω * -compact. If X is quasi M -compact and M is either finite or countable discrete in ω * , then all powers of X are countably compact. Assuming C H , we give an example of a countable subset M ω * and a quasi M -compact space X whose square is not countably compact, and show that in a model of A. Blass and S. Shelah every quasi...

Selections on Ψ -spaces

Michael HrušákPaul J. SzeptyckiArtur Hideyuki Tomita — 2001

Commentationes Mathematicae Universitatis Carolinae

We show that if 𝒜 is an uncountable AD (almost disjoint) family of subsets of ω then the space Ψ ( 𝒜 ) does not admit a continuous selection; moreover, if 𝒜 is maximal then Ψ ( 𝒜 ) does not even admit a continuous selection on pairs, answering thus questions of T. Nogura.

Extraresolvability and cardinal arithmetic

Ofelia Teresa AlasSalvador García-FerreiraArtur Hideyuki Tomita — 1999

Commentationes Mathematicae Universitatis Carolinae

Following Malykhin, we say that a space X is if X contains a family 𝒟 of dense subsets such that | 𝒟 | > Δ ( X ) and the intersection of every two elements of 𝒟 is nowhere dense, where Δ ( X ) = min { | U | : U is a nonempty open subset of X } is the of X . We show that, for every cardinal κ , there is a compact extraresolvable space of size and dispersion character 2 κ . In connection with some cardinal inequalities, we prove the equivalence of the following statements: 1) 2 κ < 2 κ + , 2) ( κ + ) κ is extraresolvable and 3) A ( κ + ) κ is extraresolvable, where A ( κ + ) ...

Topological games and product spaces

Salvador García-FerreiraR. A. González-SilvaArtur Hideyuki Tomita — 2002

Commentationes Mathematicae Universitatis Carolinae

In this paper, we deal with the product of spaces which are either 𝒢 -spaces or 𝒢 p -spaces, for some p ω * . These spaces are defined in terms of a two-person infinite game over a topological space. All countably compact spaces are 𝒢 -spaces, and every 𝒢 p -space is a 𝒢 -space, for every p ω * . We prove that if { X μ : μ < ω 1 } is a set of spaces whose product X = μ < ω 1 X μ is a 𝒢 -space, then there is A [ ω 1 ] ω such that X μ is countably compact for every μ ω 1 A . As a consequence, X ω 1 is a 𝒢 -space iff X ω 1 is countably compact, and if X 2 𝔠 is a 𝒢 -space, then all...

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