Extremal pseudocompact topological groups

Abstract

Topological groups here are assumed to satisfy the Hausdorff separation property. A topological group G is totally bounded if it embeds as a (dense) subgroup of a compact group G ¯ ; here G ¯ , the Weil completion of G, is unique in the obvious sense. It is known that every pseudocompact topological group is totally bounded; and a totally bounded group G is pseudocompact if and only if G meets each nonempty G δ -subset of G ¯ . A pseudocompact group is said to be r- extremal [resp., s- extremal] if G admits no strictly finer pseudocompact group topology [resp., G has no proper dense pseudocompact subgroup]. (Note: r- derives from “refinement'', s- from “subgroup''.) Let P be the class of non-metrizable, pseudocompact Abelian groups. The authors contribute to the growing literature (see for example J. Galindo, Sci. Math. Japonicae 55 (2001) 627) supporting the conjecture that no G ∈ P is either r- nor s-extremal—but that conjecture remains open. Except for portions of (a), the following are new results concerning G ∈ P proved here. The proofs derive largely from basic, sometimes subtle, considerations comparing the algebraic structure of G ∈ P with the algebraic structure of the Weil completion G ¯ . (a) If G is either r- or s-extremal, then r 0 ( G ) = c < w ( G ) . (b) If G is totally disconnected, then G is neither r- nor s-extremal. (c) If G is either torsion-free or countably compact, then G is not both r- and s-extremal. (d) Not every closed, pseudocompact subgroup N of G is s-extremal; if G itself is either r- or s-extremal then the witnessing N may be chosen connected. (e) If in some G ∈ P every closed subgroup in P is r-extremal, then there is connected H ∈ P with the same property. (f) If 2 ω = 2 ω 1 and every closed subgroup of G is pseudocompact, then G is not s-extremal.