Pseudocompact group topologies and totally dense subgroups

Abstract

Throughout this synopsis all topologies are Hausdorff topological group topologies, and <G, ^O is assumed compact. The symbol w denotes weight. Definition. A subgroup H of <G, JO is totally dense (in G) if HdK is dense in K for every closed subgroup K of G. We prove these results. If ^'^^ with <G, i^'> pseudocompact, then not every ^'-closed subgroup of G is ^"-closed. If w{G y ^r')> with <G, -^"> totally disconnected Abelian, then there is pseudocompact J^'gJ^. Not every infinite <(?, ^"> has a proper, totally dense subgroup. But (a) if wG, ^~ with <(?, ^O connected Abelian, or (b) if <G, ^"> is totally disconnected Abelian and in the dual group ^-primary decomposition =@ p 6 p one has \ p \ > for infinitely many primes p, then <G, ^~> has a proper, totally dense, pseudocompact subgroup. Let H be a totally dense subgroup of <G, ^~>. Then (a) |G|^2 |HI ; (b) if G is Abelian then \G\^\H\>; (c) if G is connected Abelian then |G| = |JB|; (d) if G is totally disconnected and H countably compact, then G=H; (e) there are examples with <G, JO (totally disconnected) Abelian and \H\<\G\.