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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
w⟨G,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|.
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