We show that the original concept of R-factorizability, as well as some of its modifications, examined in the realms of topological, paratopological, and semitopological groups possess an essential feature of absoluteness when transitioning to a broader category. This resolves a certain ambiguity in the research conducted to date and enables us to keep 'old' notation for formally different notions of factorizability. It is also shown that a paratopological group G is R-factorizable if and only if its T-i-reflection, T-i(G), is R-factorizable for some (equivalently, for each) i is an element of {0, 1, 2, 3}, which in turn is equivalent to the regular reflection Reg(G) of G being R-factorizable. When substituting the aforementioned reflections with the quotient group G/N, where N is the closure of the singleton {eG}, this result holds true for every topological group G. The latter results indicate a specific form of stability regarding the concept of R-factorizability. Several routes for further investigation are outlined at the end of the article.

Semitopological groups G and H are said to be HM-equivalent if the Hartman-Mycielski extensions G· and H· of G and H, respectively, are topologically isomorphic. It is shown that G and Gk are HM-equivalent, for every semitopological group G and an integer k ≥ 1 . We also show that if G and H are semitopological groups and G is topologically isomorphic to a subgroup of a finite power of H, then G· admits a topological monomorphism to H· . It is established that the HM-equivalence relation preserves a variety of properties, especially those expressed in terms of cardinal functions. On the other hand, we show under some extra set-theoretic assumptions that the cellularity, Lindelöf property, countable compactness, tightness, Fréchet-Urysohn property, etc., are not preservedby the HM-equivalence in the class of topological groups.

We continue with the study of the homogeneity properties of open subsets of topological or, more broadly, regular paratopological groups, a line of research initiated by Arhangel'skii and Choban in the frame of topological groups. Actually, our basic assumption is that an open subset O of a regular (para)topological group is homeomorphic to a paratopological group. We show that if F is a closed subset of a regular Lindel & ouml;f paratopological group G with countable cellularity and the complement O = G \ F is homeomorphic to a paratopological group, then F is the union of G(delta)-sets in G. The same conclusion holds true if G is a realcompact topological group satisfying c(G(2)) <= omega, while maintaining the same assumption regarding F. Variations of these results, with stronger conclusions, are presented. For example, if G is a regular paratopological group with Nag(G) <= omega and F subset of G is as above, then F is a zero-set in G. The same conclusion is valid if G is a topological product of regular paratopological Lindel & ouml;f Sigma-groups or an open subgroup of such a product. These results either serve to complement or extend Arhangel'skii's theorem as presented in Arhangel'skii (2023) [3]. (c) 2026 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

By a result of A.A. Markov, every Tychonoff space is embeddable as a closed subspace into a Hausdorff topological group, so there is a wealth of Hausdorff topological groups that are not normal spaces. We introduce two very wide classes of topological groups (that are not necessarily normal spaces) as follows. A Hausdorff topological group G is called N-factorizable (resp., Pc-factorizable) if for every continuous real-valued function f on G, there exists a continuous homomorphism 7r: G -> H onto a normal (resp., paracompact) topological group H such that f = h degrees 7r, for some continuous real-valued function h on H. We study the classes of N-factorizable and Pc-factorizable topological groups which contain all normal and, respectively, paracompact topological groups, in addition to all R-factorizable and M-factorizable topological groups. We show that every topological group is a quotient of a Pc-factorizable group. As it turns out, the N-factorizable groups form a proper subclass of Hausdorff topological groups, whereas the Pc-factorizable groups are a proper subclass of theN-factorizable groups. The latter two classes of groups are closed when taking perfect homomorphic images. However, similar to normal spaces, the two classes are not finitely productive, even if the factors are omega-narrow groups. Several open problems are formulated. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

For each topological group G, it is shown that the Hartman-Mycielski group G center dot over G is minimally almost periodic, that is, all continuous homomorphisms of G center dot to compact Hausdorff topological groups are trivial. We also show that the group G center dot, over a compactly generated topological group G, is algebraically generated by a compact connected subset. Consequently, a wealth of minimally almost periodic, connected, compactly generated topological Abelian groups exist. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The Lindel & ouml;f property, cellularity, countable compactness, countable pracompactness, and pseudocompactness are not finitely productive properties. Multiplying subsets of a topological group does not preserve these properties either. We continue the study started by A.V. Arhangel'skii a few years ago and show that if U is an open Lindel & ouml;f (countably cellular, or countably compact) subset of a topological group Gand a subset F of G is Lindel & ouml;f (countably cellular, count- ably compact or countably pracompact), then the group products UF and FU are also Lindel & ouml;f (countably cellular, countably compact or countably pracompact) subspaces of G. Therefore, the open subgroup of G algebraically generated by U boolean OR F is Lindel & ouml;f (countably cellular, or is the union of a countable family of open countably compact or countably pracompact subsets). Similarly, if U is an open pseudocompact subset of Gand a set F subset of G is pseudocompact, then the group products UF and FU are pseudocompact subspaces of G. It is also established that if Band Care bounded subsets of a locally feebly compact paratopological group G, then the sets B-1, C-1 and BC are bounded in G. Hence, every bounded subset of G is contained in an open sigma-bounded subgroup of G. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

An example of an infinite regular feebly compact quasitopological group is presented such that all continuous real-valued functions on the group are constant. The example delineates the boundaries for possible generalizations of Banakh–Ravsky’s theorem concerning the complete regularity of regular paratopological groups. Our construction is based on the use of Korovin orbits in X^G , where X is a special regular countably compact space constructed by Bardyla and Zdomskyy and G is an abstract Abelian group of an appropriate cardinality. Also, we study the interplay between the separation properties of the space X and Korovin orbits in X^G . We show in particular that if X contains two nonempty disjoint open subsets, then every Korovin orbit in X^G is Hausdorff. Several open problems in the field are presented.

In 1976, Nickolas showed that for each natural n, the free topological group F(X^n) is topologically isomorphic to a subgroup of F(X) provided X is a compact space or, more generally, a k_ω -space. We complement the Nickolas’ embedding theorem by showing that it remains true for every topological space X such that all finite powers of X are pseudocompact. For example, all pseudocompact k-spaces enjoy this property. Also, we extend the embedding theorem to the class of NC_ω -spaces that includes, in particular, the k_ω -spaces and the well-ordered spaces of ordinals [0, α ) , for every ordinal α . Our results are quite sharp because we present a first example of a Tychonoff space Z such that F(Z) does not contain an isomorphic copy of the group F(Z^2) . In addition, our space Z is countably compact, separable, and its square Z^2 is not pseudocompact.

We are honoured to present this Special Issue of Axioms with the title “Topology and Functional Analysis” to showcase recent work on this and related topics and to provide an opportunity for María Jesús Chasco’s friends and colleagues to pay tribute to her mathematical career on the occasion of her 65th birthday [...]

We show that all countable subsets of any pseudocompact quasitopological group in the form of a Korovin orbit are closed, discrete, and C⁎-embedded. Consequently, any infinite pseudocompact Korovin orbit is not homeomorphic to a topological group. Moreover, infinite pseudocompact Korovin orbits are not homeomorphic to any Mal'tsev space.

We prove the statement formulated in the title of the article. Then we apply it to show that there exists Lindel & ouml;f P-groups G and H satisfying w(G) = w(H) = |G| = |H| = aleph 1 such that G and H are not locally homeomorphic. This solves Problem 4.4.7 from the book (Arhangel'skii and Tkachenko, 2008 [1]) in the negative. Also, we present two homeomorphic complete Abelian P-groups one of which is omega-narrow and the other is not. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We construct a Hausdorff topological group G such that N-1 is a pre calibre of G (hence, G has countable cellularity), all countable subsets of G are closed and C-embedded in G, but G is not R-factorizable. This solves Problem 8.6.3 from the book "Topological Groups and Related Structures" (2008) in the negative.

The use of microservice architecture is a popular approach to the development of modern software systems. Microservices are used in large Internet projects serving a large number of users. Applications with microservice architecture have many advantages such as high availability, fault tolerance, flexibility, scalability. Along with the advantages, this approach also has disadvantages. In a distributed system consisting of many services, managing transactions spanning multiple services is a complex task. Since microservices are loosely coupled and the data used by a distributed transaction is stored on different nodes in the network, there is a problem of data consistency. The purpose of the article is to explore ways to manage transactions in microservices and ensure data consistency, identifying the advantages and disadvantages of applying existing design patterns. This article discusses the use of the 2PC two-phase commit protocol and the Saga design pattern. 2PC is a robust data consistency protocol for microservices. But it is a form of synchronous communication that results in microservices being coupled at runtime, which has a significant impact on the availability of the application. If high availability of the application is a priority requirement, then it is better to choose the Saga design pattern as a basis for development. It solves the problem of executing transactions spanning multiple services. To ensure data consistency, Saga is used in conjunction with the Event sourcing or Transactional Outbox pattern. Using the choreography-based saga pattern provides better service performance compared to orchestration. It is better to use it for scenarios in which response time is critical, the number of services involved in a distributed transaction is small. The orchestration-based pattern is slower than choreography, but is the best choice for handling complex transaction scenarios.

We show that for a regular P-space X, the pseudo- ℵ _1 -compactness of X is equivalent to the pseudo- ℵ _1 -compactness of the free topological group F(X) on X. This fact is applied to deduce that every regular pseudo- ℵ _1 -compact P-space is ℵ _1 -stable. Also, we calculate the exact value of the weight of F(X), for a Lindelöf P-space X, in terms of the i-weight of the space X. Using the partially ordered set , for an uncountable cardinal τ , we present a nontrivial relationship between the weight and i-weight of an arbitrary Lindelöf P-group.

We study the properties of the free abelian topological group Ad(X) on a metric space (X,d) endowed with the topology generated by the Graev extension dˆ of a given metric d on X. We find that the group of Lipschitz functions Lip0(X,T) is the group of continuous characters of Ad(X). From this fact we derive some interesting properties of the metric groups Ad(X) and Lip0(X,T).

In this article, we define a new cardinal function called the closure index of a semitopological group. The closure index ci(G) of a semitopological group G satisfying the T1-separation axiom lies between their character and pseudocharacter. For locally compact topological groups, these three cardinal functions coincide. We extend this result to the class of feathered topological groups and determine the value of the closure index for other classes of topological groups with compact-like properties. We also present several examples of topological groups with compact-like properties for which the three cardinal functions are pairwise distinct. We also find some classes of topological groups of countable pseudocharacter that have countable closure index.(c) 2022 Elsevier B.V. All rights reserved.

We present conditions on factors guaranteeing that the product of two spaces (topological groups) is (fairly) pseudo-N-1-compact or omega-stable. It is shown, for example, that the product X x Y of a regular pseudo-N-1-compact P-space X with a weakly Lindelof space Y is fairly pseudo-N-1-compact. Similarly, the product G x Y of a pseudo-N-1-compact P-group G and a fairly pseudo-N-1-compact space Y is fairly pseudo-N-1-compact. Also, we prove that for each infinite cardinal tau, a Tychonoff space X is tau-stable if and only if the free topological group F(X) (equivalently, the free Abelian topological group A(X)) on X is tau-steady. This enables us to establish T-stability of several classes of Tychonoff spaces. Additionally, we resolve Open Problem 5.6.2 from [2] by proving that the product G x H of a Lindelof Sigma-group G and a Lindelof P-group H is T-stable provided T = N-n for some n epsilon omega or T-omega = T. (c) 2022 Elsevier B.V. All rights reserved.

A topological group G with vertical bar G vertical bar > 1 is called d-independent if for every subgroup S of G with vertical bar S vertical bar < 2(omega), one can find a countable dense subgroup H of G such that S boolean AND H = {e}. Therefore, d-independent groups are separable and have cardinality at least 2(omega). Our main result is a purely algebraic characterization of d-independence in the class of compact metrizable abelian groups. We prove that a compact metrizable abelian group G with vertical bar G vertical bar > 1 is d-independent if and only if for every integer m >= 1, either vertical bar mG vertical bar = 2(omega) or vertical bar mG vertical bar = 1. This characterization implies that a compact metrizable abelian group is d-independent if and only if it is maximally fragmentable [Comfort and Dikranjan (2014) [4]] iff G an M -group as defined by Dikranjan and Shakhmatov (2016) in [7]. Also we present a characterization of separable metrizable d-independent abelian groups and show that products of separable topological groups can often be d-independent, even if the factors fail to be d-independent. (C) 2021 Elsevier B.V. All rights reserved.

Our main objective is a further study of M-factorizability in topological groups as defined in Zhang, Peng, He, Tkachenko (2020) [15]. We focus on topological-algebraic implications of M-factorizability such as tau-precompactness, pseudo-tau-compactness and tau-fineness. We also study products of topological groups and present necessary and sufficient conditions on the factors guaranteeing the M-factorizability of products. Our main technical tool for this study is the new notion of tau-fine topological group, where tau > omega is a cardinal. We prove the following dichotomy theorem: Every M-factorizable topological group is either R-factorizable or omega(1)-fine. Another dichotomy is established for the product of two groups. We prove that if the product G x H of topological groups is M-factorizable, then for every cardinal tau > omega, either G is tau-fine or H is pseudo-tau-compact. We also show that the product G x H is M-factorizable provided G is a metrizable topological group with omega(G) <= tau and H is a tau-fine topological group with hl(H) <= tau. It is also proved that the product G x H is M-factorizable (R-factorizable) whenever G is an arbitrary M-factorizable (R-factorizable) topological group and H is a locally compact separable metrizable topological group. (C) 2021 Elsevier B.V. All rights reserved.

We continue the study of M-factorizability in topological groups started in [18], with a special emphasis on feathered groups. It is shown that a feathered group G is M-factorizable if and only if G is either metrizable or R-factorizable. We also prove that an M-factorizable Čech-complete subgroup H of a topological group G is C-embedded in G. We show that the product G=∏n∈ωGn of countably many M-factorizable feathered groups is M-factorizable if and only if all the factors are metrizable or R-factorizable. We also show that continuous d-open homomorphisms preserve M-factorizability.