Subgroups and products of R-factorizable P-groups (original) (raw)

Products of R-factorizable groups

We consider the Dieudonné and Hewitt-Nachbin completions, R-factorizability, and pseudo-ℵ 1-compactness in products of spaces and topological groups in the case when one of the factors is a P-space. We prove that if X is a P-space and Y is a weakly Lindelöf space, then the formula µ(X × Y) = µX × µY holds. We also show that the product G × K of a non-discrete Rfactorizable P-group G with an R-factorizable group K is R-factorizable iff the space G × K is pseudo-ℵ 1-compact. This theorem is complemented by the fact that the product of an R-factorizable P-group with a space Y is pseudo-ℵ 1-compact provided that every locally countable family of open sets in Y is countable. As a corollary, we deduce that the product of an R-factorizable P-group with an R-factorizable weakly Lindelöf group is R-factorizable.

factorizable groups and subgroups of Lindelöf P-groups

Topology and its Applications, 2004

The main subject of our study are P -groups, that is, the topological groups whose G δ -sets are open. We establish several elementary properties of Pgroups and then prove that a P -group is R-factorizable iff it is pseudo-ω 1 -compact iff it is ω-stable. This characterization is used to show that direct products of Rfactorizable P -groups as well as continuous homomorphic images of R-factorizable P -groups are R-factorizable. A special emphasis is placed on the study of subgroups of Lindelöf P -groups.

Hereditarily -factorizable groups

Topology and its Applications, 2010

We show that every subgroup of the σ-product of a family {G i : i ∈ I} of regular paratopological groups satisfying Nag(G i) ω has countable cellularity, is perfectly κnormal and R 3-factorizable. For topological groups, we prove a more general result as follows. Let C be the minimal class of topological groups that contains all Lindelöf Σgroups and is closed under taking arbitrary subgroups, countable products, continuous homomorphic images, and forming σ-products. Then every group in C has countable cellularity, is hereditarily R-factorizable and perfectly κ-normal.

On some kinds of factorizable topological groups

Cornell University - arXiv, 2022

Based on the concepts of R-factorizable topological groups and M-factorizable topological groups, we introduce four classes of factorizabilities on topological groups, named P M-factorizabilities, P m-factorizabilities, SM-factorizabilities and P SM-factorizabilities, respectively. Some properties of the four classes of spaces are investigated.

Nondiscrete P-groups can be reflexive

Topology and its Applications, 2011

We present a series of examples of nondiscrete reflexive P-groups (i.e., groups in which all G δ-sets are open) as well as noncompact reflexive ω-bounded groups (in which the closure of every countable set is compact). Our main result implies that every product of feathered (equivalently, almost metrizable) Abelian groups equipped with the P-modified topology is a reflexive group. In particular, every compact Abelian group with the P-modified topology is reflexive. This answers a question posed by S. Hernández and P. Nickolas and solves a problem raised by Ardanza-Trevijano, Chasco, Domínguez, and Tkachenko.

Abelian pro-countable groups and orbit equivalence relations

We show, among other results, that for every non-locally compact, abelian quasi-countable group G there exists a closed subgroup L of G, and a closed, non-locally compact subgroup K of G/L which is a direct product of discrete, countable groups. As an application we prove that for every abelian Polish group G of the form H/L, where H,L are closed subgroups of Iso(X) and X is a locally compact separable metric space (e.g., G is abelian, quasi-countable), G is locally compact iff every continuous action of G on a Polish space Y induces an orbit equivalence relation that is reducible to an equivalence relation with countable classes.

Products of topological groups in which all closed subgroups are separable

Topology and its Applications

To the memory of Wistar Comfort (1933-2016), a great topologist and man, to whom we owe much of our inspiration Abstract. We prove that if H is a topological group such that all closed subgroups of H are separable, then the product G × H has the same property for every separable compact group G. Let c be the cardinality of the continuum. Assuming 2 ω 1 = c, we show that there exist: • pseudocompact topological abelian groups G and H such that all closed subgroups of G and H are separable, but the product G × H contains a closed non-separable σ-compact subgroup; • pseudocomplete locally convex vector spaces K and L such that all closed vector subspaces of K and L are separable, but the product K × L contains a closed non-separable σ-compact vector subspace.

Almost p-compact groups

Topology and its Applications, 2012

We show that if p is a selective ultrafilter, then for each cardinal α ω 1 , there exists a topological group G such that G β is almost p-compact (in particular, countably compact), for β < α, but G α is not countably compact. If in addition, we assume Martin's Axiom, then the result above holds for every α < c.

A Factorization Theorem for Topological Abelian Groups

Communications in Algebra, 2014

Using the nice properties of the w-divisible weight and the w-divisible groups, we prove a factorization theorem for compact abelian groups K; namely, K = K tor × K d , where K tor is a bounded torsion compact abelian group and K d is a w-divisible compact abelian group. By Pontryagin duality this result is equivalent to the same factorization for discrete abelian groups proved in .

A survey on reflexivity of abelian topological groups

An Abelian topological group is called strongly reflexive if every closed subgroup and every Hausdorff quotient of the group and of its dual group are reflexive. In the class of locally compact Abelian groups (LCA) there is no need to define "strong reflexivity": it does not add anything new to reflexivity, which by the Pontryagin - van Kampen Theorem is known to hold for every member of the class. In this survey we collect how much of "reflexivity" holds for different classes of groups, with especial emphasis in the classes of pseudocompact groups, omega\omegaomega-groups and PPP-groups, in which some reflexive groups have been recently detected. In section 3.5 we complete the duality relationship between the classes of PPP-groups and omega\omegaomega-bounded groups.