On refinements of ω-bounded group topologies

A topological group is ω-bounded if the closure of any countable subset is compact. Clearly, the ω-bounded groups are countably compact and hence, precompact. It has been pointed out recently that the class of ω-bounded groups is related with that of P-groups by duality (Galindo et al., 2011 [7]). In this direction, we obtain a characterization of ω-bounded topological groups by means of a property of the dual group (Theorem 2.4), and from it we deduce that a precompact group is realcompact if and only if its P-modification is complete (Theorem 3.5). Finally, we prove that for an ω-bounded group G, the next assertions are equivalent (Theorem 4.1):a) There exists an ω-bounded group topology on G strictly finer than the original.b) The dual group of G with the pointwise convergence topology is not realcompact.c) The P-modification of the dual group with the pointwise convergence topology is not complete.An important result of Comfort and van Mill establishes that for every pseudocompact Abelian topological group of uncountable weight (G,τ) there exists a pseudocompact group topology strictly finer than τ, in other words, τ is not r-extremal. In this paper we prove that the smaller class of ω-bounded groups behaves in a substantially different mode: namely, for an ω-bounded Abelian topological group there always exists a supreme ω-bounded group topology finer than the original one (Corollary 4.2). The latter plays thus the role of r-extremal in the class of ω-bounded group topologies.