Abelian torsion groups with a countably compact group topology

Abstract

a r t i c l e i n f o a b s t r a c tArticle history: Dedicated to Professor Ofélia Teresa Alas on the occasion of her 65th birthday MSC: primary 54H11, 54A25, 54A35 secondary 22A05 Comfort and Remus [W.W. Comfort, D. Remus, Abelian torsion groups with a pseudocompact group topology, Forum Math. 6 (3) (1994) 323-337] characterized algebraically the Abelian torsion groups that admit a pseudocompact group topology using the UlmKaplansky invariants. We show, under a condition weaker than the Generalized Continuum Hypothesis, that an Abelian torsion group (of any cardinality) admits a pseudocompact group topology if and only if it admits a countably compact group topology. Dikranjan and Tkachenko [D. Dikranjan, M. Tkachenko, Algebraic structure of small countably compact Abelian groups, Forum Math. 15 (6) (2003) 811-837], and Dikranjan and Shakhmatov [D. Dikranjan, D. Shakhmatov, Forcing hereditarily separable compact-like group topologies on Abelian groups, Topology Appl. 151 (1-3) (2005) 2-54] showed this equivalence for groups of cardinality not greater than 2 c .We also show, from the existence of a selective ultrafilter, that there are countably compact groups without non-trivial convergent sequences of cardinality κ ω , for any infinite cardinal κ. In particular, it is consistent that for every cardinal κ there are countably compact groups without non-trivial convergent sequences whose weight λ has countable cofinality and λ > κ.

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

“…A single selective ultrafilter was used in [3] to classify consistently all the torsion groups that admit a countable compact group topology and it may be possible to further develop the technique presented here to deal with larger free Abelian groups.…”

“…In [3], the authors showed that there are arbitrarily large countably compact groups without non-trivial convergent sequences using a single selective ultrafilter. For free Abelian groups, the authors of [8] showed that the existence of κ ω ≥ c incomparable selective ultrafilters implies the existence of a countably compact group topology on the free Abelian group of cardinality κ ω , which limitates the construction to 2 c .…”

Countably compact group topologies on arbitrarily large free Abelian groups

“…Because of the way our examples are constructed we can raise their weights in the same way as in the papers [23] or [5] and obtain the following result -the examples in these references are of I equals the Euclidean length of J, and we define the length of U as δ(U ) = δ(I). We also let δ(T) = 1.…”

Weakenings of compactness and normality on Isbell-Mrówka spaces, Hyperspaces of Vietoris and Abelian groups

“…Regarding question (2), in [17] A. Tomita and I. Castro-Pereira used some cardinal arithmetic weaker than GCH and the existence of a selective ultrafilter p to classify all the torsion groups that admit a p-compact group topology (without non-trivial convergent sequences). This gave the first arbitrarily large countably compact groups without nontrivial convergent sequences.…”