Forcing hereditarily separable compact-like group topologies on Abelian groups
The major achievement of this paper is a forcing construction of a (class of) special model(s) of ZFC in which Abelian groups of size at most \(2^{\mathfrak c}\) admit hereditarily separable group topologies with several compact-like properties and without infinite compact subsets. This is done in two steps. First, for every cardinal \(\kappa \geq \omega_2\), a new set-theoretic axiom \(\nabla_\kappa\) is introduced which implies \({\mathfrak c}=\omega_1\) and \(2^{\mathfrak c}=\kappa\). This new axiom is then applied to derive all major results of the paper ``in ZFC''. Second, the authors use forcing to prove \((\text{ConZFC} +{\mathfrak c}=\omega_1\;\&\;2^{\omega_1}=\kappa)\rightarrow \text{Con(ZFC}+\nabla_\kappa)\). In particular, \(\nabla_\kappa\) is consistent with ZFC and the power \(2^{\mathfrak c}\) of the continuum \({\mathfrak c}\) can be made ``arbitrarily large''. As a consequence, under \(\nabla_\kappa\), a characterization is accomplished of the Abelian groups \(G\) (necessarily of size at most \(2^{\mathfrak c}\)) 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 that can be made to contain no infinite compact subsets), (iv) a group topology making \(G\) into an \(S\)-space that is either precompact, or pseudocompact, or countably compact (and that can also be made without infinite compact subsets, if necessary). As a by-product, the algebraic structure of the Abelian groups of size at most \(2^{\mathfrak c}\) which possess, consistently, a countably compact group topology (without infinite compact subset, if desired) is completely described. The paper also contains several other interesting results and open questions.
scientific article; zbMATH DE number 2174903
scientific article