[Paper Review] Algebraic structure of countably compact non-torsion Abelian groups of size continuum from selective ultrafilters

Matheus Koveroff Bellini, Ana Carolina Boero+2 more|arXiv (Cornell University)|Sep 7, 2019

Advanced Topology and Set Theory25 references5 citations

TL;DR

This paper establishes a classification of non-torsion Abelian groups of size continuum that admit countably compact group topologies, using the existence of 𝔠 selective ultrafilters. It shows that such groups admit both countably compact topologies with and without non-trivial convergent sequences, and under 2𝔠 selective ultrafilters, there are 2𝔠 non-homeomorphic such topologies of any given weight in [𝔠, 2𝔠].

ABSTRACT

Assuming the existence of $\mathfrak c$ incomparable selective ultrafilters, we classify the non-torsion Abelian groups of cardinality $\mathfrak c$ that admit a countably compact group topology. We show that for each $κ\in [\mathfrak c, 2^\mathfrak c]$ each of these groups has a countably compact group topology of weight $κ$ without non-trivial convergent sequences and another that has convergent sequences. Assuming the existence of $2^\mathfrak c$ selective ultrafilters, there are at least $2^\mathfrak c$ non homeomorphic such topologies in each case and we also show that every Abelian group of cardinality at most $2^\mathfrak c$ is algebraically countably compact. We also show that it is consistent that every Abelian group of cardinality $\mathfrak c$ that admits a countably compact group topology admits a countably compact group topology without non-trivial convergent sequences whose weight has countable cofinality.

Motivation & Objective

• To classify non-torsion Abelian groups of cardinality 𝔠 that admit countably compact group topologies.
• To determine the existence and structure of such topologies with and without non-trivial convergent sequences.
• To investigate the number of non-homeomorphic countably compact group topologies realizable on these groups under selective ultrafilter assumptions.
• To explore the consistency of algebraic countable compactness for Abelian groups of size up to 2𝔠.
• To address open questions on p-compactness and ZFC realizability of countably compact free Abelian groups.

Proposed method

• Uses the existence of 𝔠 incomparable selective ultrafilters on ω to construct group topologies on non-torsion Abelian groups of size 𝔠.
• Applies techniques from selective ultrafilter-based group topology constructions, particularly leveraging p-limits of sequences.
• Employs the concept of p-limit points and ultrafilter convergence to ensure countable compactness without non-trivial convergent sequences.
• Constructs topologies on G ⊕ W_ω to transfer properties from known models, ensuring weight κ ∈ [𝔠, 2𝔠] and non-homeomorphism via ultrafilter witnesses.
• Uses the existence of 2𝔠 selective ultrafilters to generate 2𝔠 non-homeomorphic topologies by selecting ultrafilters outside the collection of ultrafilters associated with given spaces.
• Applies results from prior works on pseudocompact and countably compact group topologies, particularly those of Dikranjan, Tkachenko, and Tomita, to extend classification under stronger set-theoretic assumptions.

Experimental results

Research questions

• RQ1 Under the existence of 𝔠 selective ultrafilters, which non-torsion Abelian groups of size 𝔠 admit countably compact group topologies without non-trivial convergent sequences?
• RQ2 Can every Abelian group of size at most 2𝔠 be algebraically countably compact under the existence of 2𝔠 selective ultrafilters?
• RQ3 How many non-homeomorphic countably compact group topologies of a given weight κ ∈ [𝔠, 2𝔠] can exist on such groups?
• RQ4 Is it consistent that every Abelian group of size 𝔠 with a countably compact topology also admits one without non-trivial convergent sequences and with weight of countable cofinality?
• RQ5 Can a p-compact group topology without non-trivial convergent sequences be realized on ℤ^(𝔠) × ℚ^(𝔠) for some selective ultrafilter p?

Key findings

• Assuming the existence of 𝔠 selective ultrafilters, every non-torsion Abelian group of size 𝔠 with free rank 𝔠 and dG[n] finite or of size 𝔠 for all d|n admits a countably compact group topology without non-trivial convergent sequences.
• For each κ ∈ [𝔠, 2𝔠], such groups admit a countably compact group topology of weight κ without non-trivial convergent sequences.
• Under the assumption of 2𝔠 selective ultrafilters, there are at least 2𝔠 non-homeomorphic countably compact group topologies on each such group, both with and without non-trivial convergent sequences.
• Every Abelian group of cardinality at most 2𝔠 is algebraically countably compact under the existence of 2𝔠 selective ultrafilters.
• It is consistent that every Abelian group of size 𝔠 with a countably compact group topology admits a countably compact group topology without non-trivial convergent sequences whose weight has countable cofinality.
• The construction ensures non-homeomorphism of topologies by selecting ultrafilters outside the ultrafilter families associated with target spaces, enabling uncountable diversity of topologies.