General Topology-I

Embedding the bicyclic semigroup into countably compact topological semigroups

An infinite family ${\fancyscript{A}}\subset {\fancyscript{P}}(\omega )$ is almost disjoint (AD) if the intersection of any two distinct elements of ${\fancyscript{A}}$ is finite. We survey recent results on almost disjoint families, concentrating on their applications to topology.

Almost Disjoint Families and Topology

Recent advances in minimal topological groups

Publisher Summary This chapter discusses selected topics from the structure theory of topological groups. It contains open problems and questions covering the a number of topics including: the dimension theory of topological groups, pseudocompact and countably compact group topologies on Abelian groups, with or without nontrivial convergent sequences, categorically compact groups, sequentially complete groups, the Markov–Zariski topology, the Bohr topology, and transversal group topologies. All topological groups considered in this chapter are assumed to be Hausdorff. It is stated that Abelian group G is algebraically compact provided that an Abelian group H is found such that G × H admits a compact group topology. Algebraically compact groups form a relatively narrow subclass of Abelian groups (for example, the group ℤ of integers is not algebraically compact). On the other hand, every Abelian group G is algebraically pseudo-compact; that is, an Abelian group H can be found such that G × H ∈ P . Problems and questions related to Bohr-homeomorphic bounded Abelian groups are also discussed in the chapter.

Selected topics from the structure theory of topological groups

We prove that the existence of a selective ultrafilter on ω implies the existence of a countably compact group without non-trivial convergent sequences all of whose powers are countably compact. Hence, by using a selective ultrafilter on ω, it is possible to construct two countably compact groups without non-trivial convergent sequences whose product is not countably cornpact.

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Countably compact groups from a selective ultrafilter

We present a method to obtain countably compact group topologies that make a group without nontrivial convergent sequences from the weakest form of Martin’s Axiom, improving constructions due to van Douwen and Tkachenko. We force the existence of such topology on the free Abelian group of size 2 c , providing a partial answer to a question of Dikranjan and Shakmatov.

Forcing countably compact group topologies on a larger free abelian group

Countably compact topological group topologies on free Abelian groups from selective ultrafilters

We construct, under MAcountable, a countably compact topological subsemigroup of which is not a group, hence a counterexample for the Wallace problem. We also show that there is no p-compact counterexample for the Wallace problem, answering a question of D. Grant. Finally, we show that—in some sense—our counterexample for the Wallace problem constructed under MAcountable cannot be done in ZFC.

The Wallace problem: a counterexample from ${ m MA}sb { m countable }$ and $p$-compactness

In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number α ≤ 2c , a topological group G such that G is countably compact for all cardinals γ < α, but G is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under MAcountable. Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from MAcountable. However, the question has remained open for infinite cardinals. We show that the existence of 2c selective ultrafilters + 2c = 2 c implies a positive answer to Comfort’s question for every cardinal κ ≤ 2c . 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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Artur Hideyuki Tomita

Papers

Countably compact groups from a selective ultrafilter

Forcing countably compact group topologies on a larger free abelian group

Countably compact topological group topologies on free Abelian groups from selective ultrafilters

The Wallace problem: a counterexample from ${ m MA}sb { m countable }$ and $p$-compactness

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