Precompact groups and convergence

Abstract

We investigate precompact topological groups with topologies determined by convergent sequences (sequential and Fréchet) 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) the extra set-theoretic assumptions may be omitted (although in this case such groups are not necessarily 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.