Countably compact group topologies on the free Abelian group of size continuum (and a Wallace semigroup) from a selective ultrafilter

Abstract

To Ofelia T. Alas on the occasion of her 75th birthday.Abstract. We prove that the existence of a selective ultrafilter implies the existence of a countably compact Hausdorff group topology on the free Abelian group of size continuum. As a consequence, we show that the existence of a selective ultrafilter implies the existence of a Wallace semigroup (i.e., a countably compact both-sided cancellative topological semigroup which is not a topological group).

Countably compact group topologies on arbitrarily large free Abelian groups

“…We note explicitly that even though E T is only demanded to be maximal l.i. mod l(T ) * it will, because of item (2), depend on all of T , not just on l(T ).…”

“…The known examples were obtained in [18] under CH, in [21] under MA countable , in [17] from c incomparable selective ultrafilters and in [2] from one selective ultafilter. The last two use the known fact that a free Abelian group without non-trivial convergent sequences contains a Wallace semigroup, which was used in [18].…”

“…Tomita [22], Koszmider, Tomita and Watson [15], and Madariaga-Garcia and Tomita [17] obtained such examples using weaker assumptions. Boero, Castro Pereira and Tomita obtained such an example using a single selective ultrafilter [2]. Using 2 c selective ultrafilters, the example in [17] showed the consistency of a countably compact group topology on the free Abelian group of cardinality 2 c .…”

Countably compact group topologies on torsion-free Abelian groups

“…The idea is to use these ultrafilters to assign 𝑝-limits to enough injective sequences in ℤ (𝜅) to ensure countable compactness of the resulting topology. We take some inspiration from A. C. Boero, Castro-Pereira, et al, 2019 where a large independent family was used such that, up to a permutation, every injective sequence in ℤ (c) was part of this family.…”

“…Tomita (A. H. Tomita, 1998), Koszmider, Tomita and Watson (Koszmider et al, 2000), and Madariaga-García and Tomita (Madariaga-Garcia and A. H. Tomita, 2007) obtained such examples using weaker assumptions. Boero, Castro-Pereira and Tomita obtained such an example using a single selective ultrafilter (A. C. Boero, Castro-Pereira, et al, 2019). Using 2 c selective ultrafilters, the example in Madariaga-Garcia and A. H. Tomita, 2007 showed the consistency of a countably compact topology on the free Abelian group of cardinality 2 c .…”