Zoltán Balogh Memorial Topology Conference Abstracts Metrizably Fibered Go-spaces Forcing Hereditarily Separable Countably Compact Group Topologies

Using forcing we produce a model of ZFC + CH (with 2 arbitrarily large) and, in this model, we obtain a characterization of the Abelian groups G (necessarily of size at most 2) which admit: (i) a hereditarily separable group topology, (ii) a group topology making G into an S-space, (iii) a hereditarily separable group topology that is either precompact, or pseudocompact, or countably compact (and which can be made to contain no non-trivial convergent sequences), (iv) a hereditarily separable connected and locally connected group topology that is either pseudocompact, or countably compact (and which can be made to contain no non-trivial convergent sequences), (v) a group topology making G into an S-space that is either precompact, or pseudocompact, or countably compact (and which also can be made without non-trivial convergent sequences, if necessary). As a by-product, we completely describe the algebraic structure of the Abelian groups of size at most 2 which admit, at least consistently, a countably compact group topology (possibly without non-trivial convergent sequences). We obtain also a complete solution to a 1980 problem of van Douwen about the cofinality of the size of countably compact Abelian groups (a counter-examle to van Douwen’s conjecture was obtained recently by Tomita). Co-author: Dmitri Shakhmatov [email protected]