Countably Compact Group Research Articles

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.

The weight of a countably compact group whose cardinality has countable cofinality

We show the existence of a p-compact group whose size has countable cofinality in a forcing model. As a corollary, we show that consistently there exists a countably compact group whose size has countable cofinality and its weight is larger than the size.

Countably compact groups from a selective ultrafilter

We prove that the existence of a selective ultrafilter on ω implies the existence of a countably compact group without non-trivial convergent sequences all of whose powers are countably compact. Hence, by using a selective ultrafilter on ω, it is possible to construct two countably compact groups without non-trivial convergent sequences whose product is not countably cornpact.

A countably compact free Abelian group whose size has countable cofinality

Based on some set-theoretical observations, compactness results are given for general hit-and-miss hyperspaces. Compactness here is sometimes viewed splitting into “k-Lindelöfness” and ”k-compactness” for cardinals k. To focus only hit-and-miss structures, could look quite old-fashioned, but some importance, at least for the techniques, is given by a recent result of Som Naimpally, to who this article is hearty dedicated.

Two countably compact topological groups: One of size $\aleph _\omega $ and the other of weight $\aleph _\omega $ without non-trivial convergent sequences

E. K. van Douwen asked in 1980 whether the cardinality of a countably compact group must have uncountable cofinality in ZFC. He had shown that this was true under GCH. We answer his question in the negative. V. I. Malykhin and L. B. Shapiro showed in 1985 that under GCH the weight of a pseudocompact group without non-trivial convergent sequences cannot have countable cofinality and showed that there is a forcing model in which there exists a pseudocompact group without non-trivial convergent sequences whose weight is w 1 < c. We show that it is consistent that there exists a countably compact group without non-trivial convergent sequences whose weight is N w .

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.

Provisional solution to a Comfort–van Mill problem

We prove under the assumption of Martin's Axiom that an abstract Abelian group G of non-measurable cardinality is the intersection of countably compact subgroups of its Bohr compactification bG. This result is used to show that weakly free countably compact topological groups do not exist, thus answering a question posed by Comfort and van Mill in 1988. In fact, we show under MA that a free ( P, CC )-group over a space X exists iff X is empty, where P and CC are the classes of pseudocompact and countably compact topological groups, respectively. On the other hand, we prove the existence of a weakly free ( P, CC )-group over an arbitrary space X and show that our construction of such a group is functorial. Similar results remain valid in the Abelian case.

Independent group topologies on Abelian groups

Two non-discrete T 1 topologies τ 1, τ 2 on a set X are called independent if their intersection τ 1∩ τ 2 is the cofinite topology on X. We show that a countable group does not admit a pair of independent group topologies. We use MA to construct group topologies on the additive groups R and T independent of their usual interval topologies. These topologies have necessarily to be countably compact and cannot contain convergent sequences other than trivial. It is also proved that all proper unconditionally closed subsets of an Abelian (almost) torsion-free group are finite. Finally, we generalize the result proved for R and T by showing that every second countable group topology on an Abelian group of size 2 ω without non-trivial unconditionally closed subsets admits an independent group topology (this also requires MA). In particular, this implies that under MA, every (almost) torsion-free Abelian group of size 2 ω admits a Hausdorff countably compact group topology.

Countably compact groups satisfying the open mapping theorem

We discuss the structure and Cartesian products of the countably compact groups G that satisfy the following forms of the open mapping theorem: (a) every continuous surjective homomorphism G→H is open, (b) every continuous isomorphism G→H is open (in both cases H is a Hausdorff topological group).

Some topological groups with and some without suitable sets

If a discrete subset S of a topological group G with identity 1 generates a dense subgroup of G and S∪{1} is closed in G, then S is called a suitable set for G. We construct in ZFC a Lindelöf topological group L such that t(L)·ψ(L)≤ℵ0 and L does not have a suitable set. We also give a ZFC example of a countably compact topological group H with no suitable set; in addition, the closure of every countable subset of H is compact. It is proved that a non-pseudocompact topological group with a dense strictly σ-discrete subset has a closed suitable set. This implies, in particular, that a free (Abelian) topological group on a metrizable space has a closed suitable set.

On the number of countably compact group topologies on a free Abelian group

We show under MA(σ-centered) the existence of at least (2ω)+ non-homeomorphic topological group topologies on the free Abelian group of size 2ω which make it countably compact and separable. In particular, under GCH the maximum possible number of such topologies is attained. As a corollary, we show the existence of a semigroup which possesses (2ω)+ non-homeomorphic semigroup topologies which make it a counterexample for Wallace's Problem.

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.

Recent advances in minimal topological groups

This survey presents some recent trends and results (most of them unpublished) in minimal groups. The following are the main direction: 1. (a) permanence properties of minimal groups; 2. (b) complete minimal groups; 3. (c) countably compact minimal groups; 4. (d) algebraic structure of minimal abelian groups. In (a) we discuss preservation of minimality under the main group-theoretic operations: taking (direct or semidirect) products, quotients and (dense or closed) subgroups. Particular emphasis is given to infinite products and the critical power of minimality of a minimal abelian group G (this is the least nonminimal power of G provided such powers exist, otherwise κ( G) = 1). In particular, there exist (strongly) pseudocompact minimal abelian groups G with κ( G) = ω 1, while for countably compact minimal abelian groups G either κ( G) = 1, when the connected component of G is compact, or κ( G) = ω; otherwise. (b) and (c) are parallel but in opposite direction. In (b) we put completeness-like conditions on the minimal groups and see when they are compact. In (c) we impose countable compactness on the minimal groups and look for further conditions which may yield compactness. In this way (b) becomes a chase for precompactness, while (c) becomes a chase for completeness. This is why we dedicate in (b) special attention to the celebrated precompactness theorem for minimal abelian groups of Prodanov and Stoyanov and we offer some examples and comments in the case of nilpotent groups. Here we consider also stronger completeness conditions, as local compactness, completeness of all quotients, etc. It turns out that the question whether connected, countably compact, minimal abelian groups are compact depends on the existence of measurable cardinals. More precisely, the connected component of a countably compact minimal abelian group G must be compact whenever its size is not Ulam-measurable. In such a case κ( G) = 1 and the algebraic structure of G can be completely described. Under the assumption that there exist measurable cardinals one can construct a noncompact ω-bounded, connected, minimal abelian group G (this entails, of course, κ( G) = ω). In (d) we give an alternative exposition of the known results on this question. Our approach takes into account the connection between the algebraic invariants of the group and its topological properties. We pay special attention to the case of abelian groups of free-rank < c resolved by Schinkel (1990, Dissertation) and answer an open question of his regarding the case of torsion-free groups of large free-ranks. We show that ZFC cannot answer the question whether the free abelian group F of rank c admits minimal pseudocompact group topologies, even if F admits both (totally) minimal group topologies and pseudocompact group topologies.

Extension of functions defined on products of pseudocompact spaces and continuity of the inverse in pseudocompact groups

It is proved that a Tychonoff pseudocompact group with continuous multiplication is a topological group. It is also proved that a Tychonoff countably compact group with separately continuous multiplication is a topological group. A continuous function defined on a product X × Y of pseudocompact spaces is extended to a separately continuous function defined on the product β( X) × β( Y) of their Stone-Čhech compactifications.

On countably compact topologies on compact groups and on dyadic compacta

In this paper we apply certain classical results of S. Mazur, J. Keisler, A. Tarski and N.Th. Varopoulos to the theory of countably compact topological groups. For example, under a mild restriction on the cardinality of a compact group G, it is proved that there is no strictly stronger countably compact group topology on G. If f is a one-to-one continuous mapping of a countably compact topological group of countable tightness onto a compact Hausdorff space X, then X is metrizable. If a countably compact topological group of countable tightness acts continuously and transitively on a compact Hausdorff space X, then X is metrizable. Among the questions left unsettled is this: Let f be a sequentially continuous homomorphism of a compact topological group G onto a topological group H. Is then H countably compact? Equivalently, is then H totally bounded? Under Martin's axiom this question is answered positively.

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).

A countably compact topological group 𝐻 such that 𝐻×𝐻 is not countably compact

Using M A countable {\mathbf {M}}{{\mathbf {A}}_{{\text {countable}}}} we construct a topological group with the properties mentioned in the title.