Counting maximal topologies on countable groups and rings

Two non-discrete Hausdorff group topologies τ₁, τ₂ on a group G are called transversal if the least upper bound τ₁∨τ₂ of τ₁ and τ₂ is the discrete topology. We show that a countable group G admitting non-discrete Hausdorff group topologies admits c² pairwise transversal complete group topologies on G (so c² maximal group topologies). Moreover, every abelian group G admits ²|G|² pairwise transversal complete group topologies.     

Building suitable sets for locally compact groups by means of continuous selections

If a discrete subset S of a topological group G with the 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 apply Michael's selection theorem to offer a direct, self-contained, purely topological proof of the result of Hofmann and Morris [K.-H. Hofmann, S.A. Morris, Weight and c, J. Pure Appl. Algebra 68 (1-2) (1990) 181-194] on the existence of suitable sets in locally compact groups. Our approach uses only elementary facts from (topological) group theory.     

CLP-compactness for topological spaces and groups

We study CLP-compact spaces (every cover consisting of clopen sets has a finite subcover) and CLP-compact topological groups. In particular, we extend a theorem on CLP-compactness of products from [J. Steprāns, A. Šostak, Restricted compactness properties and their preservation under products, Topology Appl. 101 (3) (2000) 213-229] and we offer various criteria for CLP-compactness for spaces and topological groups, that work particularly well for precompact groups. This allows us to show that arbitrary products of CLP-compact pseudocompact groups are CLP-compact. For every natural n we construct:

(i)

a totally disconnected, n-dimensional, pseudocompact CLP-compact group; and

(ii)

a hereditarily disconnected, n-dimensional, totally minimal, CLP-compact group that can be chosen to be either separable metrizable or pseudocompact (a Hausdorff group G is totally minimal when all continuous surjective homomorphisms G→H, with a Hausdorff group H, are open).

     

Limits in compact Abelian groups

For X a compact Abelian group and B an infinite subset of its dual , let CB be the set of all x∈X such that converges to 1. If F is a free filter on , let . The sets CB and DF are subgroups of X. CB always has Haar measure 0, while the measure of DF depends on F. We show that there is a filter F such that DF has measure 0 but is not contained in any CB. This generalizes previous results for the special case where X is the circle group.     

Classes defined by stars and neighbourhood assignments

We apply and develop an idea of E. van Douwen used to define D-spaces. Given a topological property P, the class P^∗ dual to P (with respect to neighbourhood assignments) consists of spaces X such that for any neighbourhood assignment there is Y⊂X with Y∈P and . We prove that the classes of compact, countably compact and pseudocompact are self-dual with respect to neighbourhood assignments. It is also established that all spaces dual to hereditarily Lindelöf spaces are Lindelöf. In the second part of this paper we study some non-trivial classes of pseudocompact spaces defined in an analogous way using stars of open covers instead of neighbourhood assignments.     

Continuity in topological groups

We prove that, when G is a group equipped with a Baire and metrizable topology, if there is a second category dense subset S of G such that the right translations ρs and ρs⁻¹ are continuous for all s∈S and each left translation λs, s∈G, is almost-continuous (defined below) on a residual subset of G, then G is a topological group. Among other consequences, this yields that when G is a second countable locally compact right topological group, its topological centre is a topological group.     

Minimal pseudocompact group topologies on free abelian groups

A Hausdorff topological group G is minimal if every continuous isomorphism f:G→H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov, we prove the following theorem: For every infinite minimal abelian group G there exists a sequence of cardinals such that

     

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

Let G be compact abelian group such that w(C(G))=w(Cω(G)). We prove that if

|C(G)|?m(G/C(G)),     

Bounded sets in topological groups and embeddings

We show that the existence of a non-metrizable compact subspace of a topological group G often implies that G contains an uncountable supersequence (a copy of the one-point compactification of an uncountable discrete space). The existence of uncountable supersequences in a topological group has a strong impact on bounded subsets of the group. For example, if a topological group G contains an uncountable supersequence and K is a closed bounded subset of G which does not contain uncountable supersequences, then any subset A of K is bounded in G?(K?A). We also show that every precompact Abelian topological group H can be embedded as a closed subgroup into a precompact Abelian topological group G such that H is bounded in G and all bounded subsets of the quotient group G/H are finite. This complements Ursul's result on closed embeddings of precompact groups to pseudocompact groups.     

Topologies on groups determined by discrete subsets

Let G be an infinite group. Given a filter F on G, let T[F] denote the largest left invariant topology on G in which F converges to the identity. In this paper, we study the topology T[F] in case when F contains the Fréchet filter and there is such that all the subsets xM(x), where x∈G, are pairwise disjoint. We show that T[F] possesses interesting extremal properties. We consider also the question whether T[F] can be a group topology.     

Self-duality in the class of precompact groups

A topological Abelian group G is called (strongly) self-dual if there exists a topological isomorphism Φ:G→G^∧ of G onto the dual group G^∧ (such that Φ(x)(y)=Φ(y)(x) for all x,y∈G). We prove that every countably compact self-dual Abelian group is finite. It turns out, however, that for every infinite cardinal κ with κω=κ, there exists a pseudocompact, non-compact, strongly self-dual Boolean group of cardinality κ.     

The three space problem in topological groups

We study compact, countably compact, pseudocompact, and functionally bounded sets in extensions of topological groups. A property P is said to be a three space property if, for every topological group G and a closed invariant subgroup N of G, the fact that both groups N and G/N have P implies that G also has P. It is shown that if all compact (countably compact) subsets of the groups N and G/N are metrizable, then G has the same property. However, the result cannot be extended to pseudocompact subsets, a counterexample exists under p=c. Another example shows that extensions of groups do not preserve the classes of realcompact, Dieudonné complete and μ-spaces: one can find a pseudocompact, non-compact Abelian topological group G and an infinite, closed, realcompact subgroup N of G such that G/N is compact and all functionally bounded subsets of N are finite. Several examples given in the article destroy a number of tempting conjectures about extensions of topological groups.     

Factorization properties of paratopological groups

In this article we continue the study of R$\mathbb{R}$-factorizability in paratopological groups. It is shown that: (1) all concepts of R$\mathbb{R}$-factorizability in paratopological groups coincide; (2) a Tychonoff paratopological group G   is R$\mathbb{R}$-factorizable if and only if it is totally ω  -narrow and has property ω-QU$\omega \text{-}\mathit{QU}$; (3) every subgroup of a _T1$T_1$ paratopological group G   is R$\mathbb{R}$-factorizable provided that the topological group G^?$G^{?}$ associated to G is a Lindelöf Σ-space, i.e., G is a totally Lindelöf Σ-space  ; (4) if Π=_∏i∈IGi$\Pi ={\prod }_{i\in I}{G}_i$ is a product of _T1$T_1$ paratopological groups which are totally Lindelöf Σ-spaces, then each dense subgroup of Π   is R$\mathbb{R}$-factorizable. These results answer in the affirmative several questions posed earlier by M. Sanchis and M. Tkachenko and by S. Lin and L.-H. Xie.     

On subgroups of minimal topological groups

A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. The Roelcke uniformity (or lower uniformity) on a topological group is the greatest lower bound of the left and right uniformities. A group is Roelcke-precompact if it is precompact with respect to the Roelcke uniformity. Many naturally arising non-Abelian topological groups are Roelcke-precompact and hence have a natural compactification. We use such compactifications to prove that some groups of isometries are minimal. In particular, if U₁ is the Urysohn universal metric space of diameter 1, the group Iso(U₁) of all self-isometries of U₁ is Roelcke-precompact, topologically simple and minimal. We also show that every topological group is a subgroup of a minimal topologically simple Roelcke-precompact group of the form Iso(M), where M is an appropriate non-separable version of the Urysohn space.     

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.     

HFD groups in the Solovay model

Hajnal and Juhász proved that under CH there is a hereditarily separable, hereditarily normal topological group without non-trivial convergent sequences that is countably compact and not Lindelöf. The example constructed is a topological subgroup H⊆ω₁² that is an HFD with the following property

(P)

the projection of H onto every partial product I² for I∈ω[ω₁] is onto.

Any such group has the necessary properties. We prove that if κ is a cardinal of uncountable cofinality, then in the model obtained by forcing over a model of CH with the measure algebra on κ², there is an HFD topological group in ω₁² which has property (P).     

On the supremum of the pseudocompact group topologies

P is the class of pseudocompact Hausdorff topological groups, and P^′ is the class of groups which admit a topology T such that (G,T)∈P. It is known that every G=(G,T)∈P is totally bounded, so for G∈P^′ the supremum T^∨(G) of all pseudocompact group topologies on G and the supremum T#(G) of all totally bounded group topologies on G satisfy T^∨⊆T#.The authors conjecture for abelian G∈P^′ that T^∨=T#. That equality is established here for abelian G∈P^′ with any of these (overlapping) properties. (a) G is a torsion group; (b) |G|?c²; (c) r₀(G)=|G|=ω|G|; (d) |G| is a strong limit cardinal, and r₀(G)=|G|; (e) some topology T with (G,T)∈P satisfies w(G,T)?c; (f) some pseudocompact group topology on G is metrizable; (g) G admits a compact group topology, and r₀(G)=|G|. Furthermore, the product of finitely many abelian G∈P^′, each with the property T^∨(G)=T#(G), has the same property.