Abelian torsion groups with a countably compact group topology

摘要

Comfort and Remus [W.W. Comfort, D Remus, Abelian torsion groups with a pseudo-compact group topology, Forum Math. 6 (3) (1994) 323-337] characterized algebraically the Abelian torsion groups that admit a pseudocompact group topology using the Ulm-Kaplansky invariants.rnWe 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. Dikran-jan, M. Tkachenko, Algebraic structure of small countably compact Abelian groups, Forum Math. 15 (6) (2003) 811-837], and Dikranjan and Shakhmatov [D. Dikranjan, D. Shakhma-tov, 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.rnWe also show, from the existence of a selective ultrafilter, that there are countably compact groups without non-trivial convergent sequences of cardinality k~ω, for any infinite cardinal k. In particular, it is consistent that for every cardinal k there are countably compact groups without non-trivial convergent sequences whose weight λ has countable cofinality and λ > k.