Pseudocompact Subgroup Research Articles

Non-Abelian Pseudocompact Groups

Here are three recently-established theorems from the literature. (A) (2006) Every non-metrizable compact abelian group K has 2|K| -many proper dense pseudocompact subgroups. (B) (2003) Every non-metrizable compact abelian group K admits 22|K| -many strictly finer pseudocompact topological group refinements. (C) (2007) Every non-metrizable pseudocompact abelian group has a proper dense pseudocompact subgroup and a strictly finer pseudocompact topological group refinement. (Theorems (A), (B) and (C) become false if the non-metrizable hypothesis is omitted.) With a detailed view toward the relevant literature, the present authors ask: What happens to (A), (B), (C) and to similar known facts about pseudocompact abelian groups if the abelian hypothesis is omitted? Are the resulting statements true, false, true under certain natural additional hypotheses, etc.? Several new results responding in part to these questions are given, and several specific additional questions are posed.

Precompact noncompact reflexive Abelian groups

We study Pontryagin reflexivity in the class of precompact topological Abelian groups. We find reflexive groups among precompact not pseudocompact and among pseudocompact not compact groups. Making use of Martin's axiom we give an example of a reflexive countably compact not compact Abelian group. We also prove that every pseudocompact Abelian group is a quotient of a reflexive pseudocompact group with respect to a closed reflexive pseudocompact subgroup.

Dense minimal pseudocompact subgroups of compact Abelian groups

Motivated by a recent theorem of Comfort and van Mill, we study when a pseudocompact Abelian group admits proper dense minimal pseudocompact subgroups and give a complete answer in the case of compact Abelian groups. Moreover we characterize compact Abelian groups that admit proper dense minimal subgroups.

Pseudocompact totally dense subgroups

It was shown by Dikranjan and Shakhmatov in 1992 that if a compact abelian group K K admits a proper totally dense pseudocompact subgroup, then K K cannot have a torsion closed G δ G_\delta -subgroup; moreover this condition was shown to be also sufficient under LH. We prove in ZFC that this condition actually ensures the existence of a proper totally dense subgroup H H of K K that contains an ω \omega -bounded dense subgroup of K K (such an H H is necessarily pseudocompact). This answers two questions posed by Dikranjan and Shakhmatov (Proc. Amer. Math. Soc. 114 (1992), 1119–1129).

Pseudocompact groups: progress and problems

Several months ago the speaker and Jan van Mill gave a proof of this result [W.W. Comfort, J. van Mill, Extremal pseudocompact abelian groups are compact metrizable, Abstracts Amer. Math. Soc. 27 (2006) 78 (Abstract #1014-22-958); W.W. Comfort, J. van Mill, Extremal pseudocompact abelian groups are compact metrizable, Proc. Amer. Math. Soc. 135 (2007) 4039–4044]: A pseudocompact abelian group of uncountable weight admits both a proper dense pseudocompact subgroup and a strictly larger pseudocompact group topology. This presentation will describe both the necessary new details of the argument and the historical development (useful tools, special cases). Among those who contributed essentially are: K.A. Ross (1964, 1966); T. Soundararajan (1982); L.C. Robertson (1982, 1988); J. van Mill (1989); J. van Mill and H. Gladdines (1994); J. Galindo (2002). Several related unsolved problems will be cited.

Extremal pseudocompact Abelian groups are compact metrizable

Every pseudocompact Abelian group of uncountable weight has both a proper dense pseudocompact subgroup and a strictly finer pseudocompact group topology.

Compact groups containing dense pseudocompact subgroups without non-trivial convergent sequences

Let G be compact abelian group such that w ( C ( G ) ) = w ( C ( G ) ) ω . We prove that if | C ( G ) | ⩾ m ( G / C ( G ) ) , then G contains a dense pseudocompact subgroup without non-trivial convergent sequences, where C ( G ) is the component of the identity of G and m ( G ) is the smallest cardinality of a dense pseudocompact subgroup of G. As a consequence we obtain the following: (1) Every compact connected abelian group of weight κ with κ = κ ω has a dense pseudocompact subgroup without non-trivial convergent sequences. (2) [GCH] Let G be a compact abelian group whose connected component has weight κ with κ = κ ω . The following assertions are then equivalent: (i) Every dense pseudocompact subgroup of G has a non-trivial convergent sequence. (ii) One of the following two conditions is satisfied: (a) For some n < ω , nG is infinite and cf ( w ( n G ) ) = ω . (b) | C ( G ) | < m ( log ( r 0 ( G / C ( G ) ) ) ) .

Weakly metrizable pseudocompact groups

We study various weaker versions of metrizability for pseudocompact abelian groups G: singularity (G possesses a compact metrizable subgroup of the form mG, m > 0), almost connectedness (G is metrizable modulo the connected component) and various versions of extremality in the sense of Comfort and co-authors (s-extremal, if G has no proper dense pseudocompact subgroups, r-extremal, if G admits no proper pseudocompact refinement). We introduce also weaklyextremal pseudocompact groups (weakening simultaneously s-extremal and r-extremal). It turns out that this “symmetric” version of ex-tremality has nice properties that restore the symmetry, to a certain extent, in the theory of extremal pseudocompact groups obtaining simpler uniform proofs of most of the known results. We characterize doubly extremal pseudocompact groups within the class of s-extremal pseudocompact groups. We give also a criterion for r-extremality for connected pseudocompact groups.

Extremal pseudocompact topological groups

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.

Cardinal numbers associated with dense pseudocompact, countably compact, and ω-bounded subgroups

A brief survey is offered in Section 1 on earlier progress in the solution of two related problems concerning dense proper pseudocompact (countably compact, ω-bounded) subgroups of compact nonmetrizable groups. These are: Does every nonmetrizable compact group contain such a subgroup and if a compact group has such a subgroup how large may a distinguished family of such subgroups be? Section 2 contains new results and considers the second question in detail. It is shown that each compact nonmetrizable group G that is product-like contains a family of 2 ¦G¦ distinct dense pseudocompact subgroups. In the special case where L is a Cartesian product of more than ω 1 compact simply connected simple Lie groups, L even contains 2 ¦L¦ free subgroups that are dense and pseudocompact. In the final section, it is shown that if 2 ω < 2 ω 1 then each nonmetrizable compact group contains at least 2 ω 1 distinct dense countably compact subgroups. Conditions are given under which a compact group has a large collection of dense ω-bounded subgroups. Finally, this section gives an example due to Comfort, poses some additional questions, and records a recent development.

Large families of dense pseudocompact subgroups of compact groups

Large families of dense pseudocompact subgroups of compact groups

Proper Pseudocompact Subgroups of Pseudocompact Abelian Groups

ABSTRACT: We prove among other things that if G is a pseudocompact Abelian topological group such that |G| &gt; c or ω1≤w(G)≤ c then G has a proper dense pseudocompact subgroup.

Compact-Like Totally Dense Subgroups of Compact Groups

A subgroup $H$ of a topological group $G$ is (weakly) totally dense in $G$ if for each closed (normal) subgroup $N$ of $G$ the set $H \cap N$ is dense in $N$. We show that no compact (or more generally, $\omega$-bounded) group contains a proper, totally dense, countably compact subgroup. This yields that a countably compact Abelian group $G$ is compact if and only if each continuous homomorphism $\pi :G \to H$ of $G$ onto a topological group $H$ 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 ${G_\delta }$-subgroup. If a topological group contains a proper, totally dense, pseudocompact subgroup, then none of its closed, normal ${G_\delta }$-subgroups is torsion. Under Lusin’s hypothesis ${2^{{\omega _1}}} = {2^\omega }$ the converse is true for a compact Abelian group $G$. If $G$ is a compact Abelian group with nonmetrizable connected component of zero, then there are a dense, countably compact subgroup $K$ of $G$ and a proper, totally dense subgroup $H$ of $G$ with $K \subseteq H$ (in particular, $H$ is pseudocompact).

Compact-like totally dense subgroups of compact groups

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).

Concerning connected, pseudocompact Abelian groups

It is known that if P is either the property ω-bounded or countably compact, then for every cardinal α ⩾ ω there is a P-group G such that wG = α and no proper, dense subgroup of G is a P-group. What happens when P is the property pseudocompact? The first-listed author and Robertson have shown that every zero-dimensional Abelian P-group G with wG > ω has a proper, dense, P-group. Turning to the case of connected P-groups, the present authors show the following results: Let G be a connected, pseudocompact, Abelian group with wG = α > ω. If any one of the following conditions holds, then G has a proper, dense (necessarily connected) pseudocompact subgroup: (a) wG ⩽ c ; (b) | G| ⩾ α ω ; (c) α is a strong limit cardinal and cf( α) > ω; (d) | torG| > c (e) G is not divisible.

Pseudocompact group topologies and totally dense subgroups

Throughout this synopsis all topologies are Hausdorff topological group topologies, and pseudocompact, then not every ^'-closed subgroup of G is ^-closed. If w{Gy^r')>ω with totally disconnected Abelian, then there is pseudocompact J^'gJ^. Not every infinite has a proper, totally dense subgroup. But (a) if wζG, ^~»ω with is totally disconnected Abelian and in the dual group ^-primary decomposition ό=@p6p one has \όp >ω for infinitely many primes p, then has a proper, totally dense, pseudocompact subgroup. Let H be a totally dense subgroup of . Then (a) |G|^2|HI; (b) if G is Abelian then \G\^\H\°>; (c) if G is connected Abelian then |G| = |JBΓ|; (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\.

Dense Subgroups of Compact Groups

A new short proof is given showing that infinite compact groups contain dense pseudocompact subgroups of small cardinality. Relationships between connectedness and divisibility in pseudocompact groups are given. It is shown that most compact connected Abelian groups contain proper dense connected subgroups.

Dense subgroups of compact groups

A new short proof is given showing that infinite compact groups contain dense pseudocompact subgroups of small cardinality. Relationships between connectedness and divisibility in pseudocompact groups are given. It is shown that most compact connected Abelian groups contain proper dense connected subgroups.