Precompact groups and convergence

Abstract

We consider precompact sequential and Fréchet group topologies and show that some natural constructions of such topologies always result in metrizable groups answering a question of D. Dikranjan et al. We show that it is consistent that all sequential precompact topologies on countable groups are Fréchet (or even metrizable). For some classes of groups (for example boolean) extra set-theoretic assumptions may be omitted (although in this case such groups do not have to be metrizable).We also build (using ♦) an example of a countably compact Fréchet group that is not α3 and obtain a counterexample to a conjecture of D. Shakhmatov as a corollary.

Invariant ideal axiom, beyond the countable sequential groups

“…The final result of this section answers [25,Question 7.4] by showing that IIA implies the nonexistence of sequential precompact groups that are not Fréchet. For countable sequential groups in the Cohen model this was established in [32]. It was shown in [26] that countably compact sequential non-Fréchet groups exist under ♢, which together with the result below implies that the existence of precompact sequential non-Fréchet groups is independent of the usual axioms of ZFC.…”