Vol. 100, No. 1, 1982

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Pseudocompact group topologies and totally dense subgroups

W. Wistar (William) Comfort and T. Soundararajan

Vol. 100 (1982), No. 1, 61–84

Throughout this synopsis all topologies are Hausdorff topological group topologies, and G,𝒯⟩ is assumed compact. The symbol w denotes weight. Definition. A subgroup H of G,𝒯⟩ is totally dense (in G) if H K is dense in K for every closed subgroup K of G. We prove these results.

If 𝒯′ 𝒯 with G,𝒯′⟩ pseudocompact, then not every 𝒯′-closed subgroup of G is 𝒯 -closed. If w(G,𝒯 ) > ω with G,𝒯⟩ totally disconnected Abelian, then there is pseudocompact 𝒯′ 𝒯 .

Not every infinite G,𝒯⟩ has a proper, totally dense subgroup. But (a) if wG,𝒯⟩ > ω with G,𝒯⟩ connected Abelian, or (b) if G,𝒯⟩ is totally disconnected Abelian and in the dual group p-primary decomposition Ĝ = pĜp one has |Ĝp| > ω for infinitely many primes p, then G,𝒯⟩ has a proper, totally dense, pseudocompact subgroup.

Let H be a totally dense subgroup of G,𝒯⟩. Then (a) |G|2|H|; (b) if G is Abelian then |G||H|ω; (c) if G is connected Abelian then |G| = |H|; (d) if G is totally disconnected and H countably compact, then G = H; (e) there are examples with G,𝒯⟩ (totally disconnected) Abelian and |H| < |G|.

Mathematical Subject Classification 2000
Primary: 22B05
Secondary: 54A10
Received: 18 February 1981
Revised: 27 May 1981
Published: 1 May 1982
W. Wistar (William) Comfort
T. Soundararajan