Countably Compact Abelian Groups Research Articles

Density character of subgroups of topological groups

We give a complete characterization of subgroups of separable topological groups. Then we show that the following conditions are equivalent for an ω \omega -narrow topological group G G : (i) G G is homeomorphic to a subspace of a separable regular space; (ii) G G is topologically isomorphic to a subgroup of a separable topological group; (iii) G G is topologically isomorphic to a closed subgroup of a separable path-connected, locally path-connected topological group. A pro-Lie group is a projective limit of finite-dimensional Lie groups. We prove here that an almost connected pro-Lie group is separable if and only if its weight is not greater than the cardinality c \mathfrak {c} of the continuum. It is deduced from this that an almost connected pro-Lie group is separable if and only if it is homeomorphic to a subspace of a separable Hausdorff space. It is also proved that a locally compact (even feathered) topological group G G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G G is homeomorphic to a subspace of a separable Tychonoff space. We show that every precompact (abelian) topological group of weight less than or equal to c \mathfrak {c} is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight c \mathfrak {c} . This result implies that there is a wealth of closed non-separable subgroups of separable pseudocompact groups. An example is also presented under the Continuum Hypothesis of a separable countably compact abelian group which contains a non-separable closed subgroup.

Abelian torsion groups with a countably compact group topology

Comfort and Remus [W.W. Comfort, D. Remus, Abelian torsion groups with a pseudocompact group topology, Forum Math. 6 (3) (1994) 323–337] characterized algebraically the Abelian torsion groups that admit a pseudocompact group topology using the Ulm–Kaplansky invariants. We show, under a condition weaker than the Generalized Continuum Hypothesis, that an Abelian torsion group (of any cardinality) admits a pseudocompact group topology if and only if it admits a countably compact group topology. Dikranjan and Tkachenko [D. Dikranjan, M. Tkachenko, Algebraic structure of small countably compact Abelian groups, Forum Math. 15 (6) (2003) 811–837], and Dikranjan and Shakhmatov [D. Dikranjan, D. Shakhmatov, Forcing hereditarily separable compact-like group topologies on Abelian groups, Topology Appl. 151 (1–3) (2005) 2–54] showed this equivalence for groups of cardinality not greater than 2 c . We also show, from the existence of a selective ultrafilter, that there are countably compact groups without non-trivial convergent sequences of cardinality κ ω , for any infinite cardinal κ. In particular, it is consistent that for every cardinal κ there are countably compact groups without non-trivial convergent sequences whose weight λ has countable cofinality and λ > κ .

Square of countably compact groups without non-trivial convergent sequences

We show in ZFC that the existence of a countably compact Abelian group without non-trivial convergent sequences implies the existence of a countably compact group whose square is not countably compact. This improves a result obtained by van Douwen in 1980: the existence of a countably compact Boolean group without non-trivial convergent sequences implies the existence of two countably compact groups whose product is not countably compact in ZFC. Hart and van Mill showed in 1991 the existence of a countably compact group whose square is not countably compact under Martin's Axiom for countable posets. We show that the existence of such an example does not depend on some form of Martin's Axiom.

Forcing hereditarily separable compact-like group topologies on Abelian groups

Let c denote the cardinality of the continuum. Using forcing we produce a model of ZFC + CH with 2 c “arbitrarily large” and, in this model, obtain a characterization of the Abelian groups G (necessarily of size at most 2 c ) which admit: (i) a hereditarily separable group topology, (ii) a group topology making G into an S-space, (iii) a hereditarily separable group topology that is either precompact, or pseudocompact, or countably compact (and which can be made to contain no infinite compact subsets), (iv) a group topology making G into an S-space that is either precompact, or pseudocompact, or countably compact (and which also can be made without infinite compact subsets if necessary). As a by-product, we completely describe the algebraic structure of the Abelian groups of size at most 2 c which possess, at least consistently, a countably compact group topology (without infinite compact subsets, if desired). We also put to rest a 1980 problem of van Douwen about the cofinality of the size of countably compact Abelian groups.

Algebraic structure of small countably compact Abelian groups

Under Martin’s Axiom, we completely characterize the algebraic structure of Abelian groups of the size c that admit a countably compact Hausdor group topology. It turns out that, in the torsion case, these are exactly the groups that admit a pseudocompact Hausdor group topology, but this is no more valid for non-torsion Abelian groups. The algebraic constraints for the existence of a countably compact Hausdor group topology on an Abelian group G of size c are relatively simple: for every integer n, the subgroup G[n]=fx2G:nx=0g of G has to be nite or satisfy jG[n]j=c. The same has to hold true for the subgroup dG[n], where d is any divisor of n. In addition, if G is non-torsion, then the free rank r(G) of G must be equal to c.

Varieties generated by countably compact Abelian groups

We prove under the assumption of Martin's Axiom that every precompact Abelian group of size ≤ 2 N 0 belongs to the smallest class of groups that contains all Abelian countably compact groups and is closed under direct products, taking closed subgroups and continuous isomorphic images.

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

We describe the structure of totally disconnected minimal ω \omega - bounded abelian groups by reducing the description to the case of those of them which are subgroups of powers of the p p -adic integers Z p \mathbb {Z}_{p} . In this case the description is obtained by means of a functorial correspondence, based on Pontryagin duality, between topological and linearly topologized groups introduced by Tonolo. As an application we answer the question (posed in Pseudocompact and countably compact abelian groups: Cartesian products and minimality, Trans. Amer. Math. Soc. 335 (1993), 775–790) when arbitrary powers of minimal ω \omega -bounded abelian groups are minimal. We prove that the positive answer to this question is equivalent to non-existence of measurable cardinals.

Pseudocompact and Countably Compact Abelian Groups: Cartesian Products and Minimality

Pseudocompact and Countably Compact Abelian Groups: Cartesian Products and Minimality

Pseudocompact and countably compact abelian groups: Cartesian products and minimality

Denote by G the class of all Abelian Hausdorff topological groups. A group G ∈ G is minimal (totally minimal) if every continuous group isomorphism (homomorphism) i: G → H of G onto H ∈ G is open. For G ∈ G let κ(G) be the smallest cardinal τ ≥ 1 such that the minimality of G τ implies the minimality of all powers of G. For Q ⊂ G, Q¬=O, we set κ(Q) = sup{κ(G): G ∈ G} and denote by α(Q) the smallest cardinal τ ≥ 1 having the following property: If {G i : i ∈ I} ⊂ Q, I ¬= O, and each subproduct Π{G i : i ∈ J}, with J ⊂ I, J ¬= O, and |J| ≤ τ, is minimal, then the whole product Π{G i : i ∈ I} is minimal

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).