Compact-like totally dense subgroups of compact groups

Abstract

A subgroup H H of a topological group G G is (weakly) totally dense in G G if for each closed (normal) subgroup N N of G G the set H ∩ N H \cap N is dense in N N . We show that no compact (or more generally, ω \omega -bounded) group contains a proper, totally dense, countably compact subgroup. This yields that a countably compact Abelian group G G is compact if and only if each continuous homomorphism π : G → H \pi :G \to H of G G onto a topological group H H is open. Here "Abelian" cannot be dropped. A connected, compact group contains a proper, weakly totally dense, countably compact subgroup if and only if its center is not a G δ {G_\delta } -subgroup. If a topological group contains a proper, totally dense, pseudocompact subgroup, then none of its closed, normal G δ {G_\delta } -subgroups is torsion. Under Lusin’s hypothesis 2 ω 1 = 2 ω {2^{{\omega _1}}} = {2^\omega } the converse is true for a compact Abelian group G G . If G G is a compact Abelian group with nonmetrizable connected component of zero, then there are a dense, countably compact subgroup K K of G G and a proper, totally dense subgroup H H of G G with K ⊆ H K \subseteq H (in particular, H H is pseudocompact).

Pseudocompact totally dense subgroups

“…The proof of Theorem 1.5 is quite different from that of Theorem 1.4 given in [9]. In fact, following [5] we add a third condition, that is, K has the property T D ω , which seems to be stronger than the others, but it turns out to be equivalent to them and it helps to prove the theorem in its full generality.…”

“…To prove the other one in [9] the authors first proved it in case the group K has cardinality c. In particular, under the assumption of LH, the implication holds true for groups K of weight ω 1 (as |K| = 2 ω 1 = c). In the general case, supposing that K has no torsion, closed G δ -subgroups, they constructed a continuous surjective homomorphism from K to a compact abelian group H of weight ω 1 with the same property.…”

“…In other words the totally dense subgroups of a compact abelian group K have a compactness-like property. Therefore it is natural to expect that in the presence of other compactness-like properties they may turn out to be actually compact, hence coincide with K. Indeed no compact abelian group K has a proper totally dense countably compact subgroup [9,Theorem 1.4]. We recall here that a topological group G is countably compact if every countable open cover of G has a finite subcover, G is pseudocompact if every continuous real-valued function of G is bounded, and G is ω-bounded if every countable subset of G is contained in a compact subgroup of G. The following chain of implications holds:…”

Non-Abelian Pseudocompact Groups

“…Since this topic is a bit removed from our central focus here, for details in this direction we simply refer the reader to the relevant papers known to us: [50,51], [36] [52][53][54][55]. We note explicitly that, building upon and extending results from her thesis [56], Giordano Bruno and Dikranjan [57] characterized those compact abelian groups with a proper totally dense pseudocompact subgroup as those with no closed torsion G δ -subgroup.…”

On the minimality of powers of minimal 𝜔-bounded abelian groups

“…An example of a totally disconnected minimal, ω-bounded non-compact abelian group was given in [DS3,Theorem 1.5]. A connected example was given in [DS2]; it was again non-abelian. In fact, item (b) of the following theorem from [D4] shows that the connected minimal countably compact abelian groups are "frequently" compact.…”