Topological Groups (Closed)

A topical collection in Axioms (ISSN 2075-1680). This collection belongs to the section "Geometry and Topology".

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1. School of Engineering, IT and Physical Sciences, Federation University Australia, Ballarat, VIC 3353, Australia
2. Department of Mathematical and Physical Sciences, La Trobe University, Melbourne, VIC 3086, Australia
Interests: topological groups; especially locally compact groups; pro-Lie groups; topological algebra; topological vector spaces; Banach spaces; topology; group theory; functional analysis; universal algebra; transcendental number theory; numerical geometry; history of mathematics; information technology security; health informatics; international education; university education; online education; social media in the teaching of mathematics; stock market prediction; managing scholarly journals
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Topical Collection Information

Dear Colleagues,

For over a century, topological groups have been an active area of research. In 1900, David Hilbert presented a seminal address to the International Congress of Mathematician, in which he formulated 23 problems that influenced a vast amount of research of the 20th century. The fifth of these problems, Hilbert5, asked whether every locally euclidean topological group admits a Lie group structure and this motivated an enormous effort on locally compact groups. It culminated in the work of Gleason, Iwasawa, Montgomery, Yamabe, and Zippin, yielding a positive answer to Hilbert5 and important structure theory of locally compact groups. Later, Jean Dieudonné quipped that Lie groups had moved to the centre of mathematics and that one cannot undertake anything without them. A modern introduction to Lie Groups is given in the book by Hilgert and Neeb. Recently there has been much interest in infinite-dimensional Lie groups including significant publications by Glöckner and Neeb, and two books by Hofmann and Morris, which demonstrated the power of Lie Theory in describing the structure of compact groups and (almost) connected pro-Lie Groups. Advances in profinite group theory are described in books by Wilson and by Ribes and Zaleskii and on locally compact totally disconnected groups in the papers of Willis and collaborators. Over some decades the Moscow school led by Arhangel’skii produced many beautiful results on free topological groups and non-locally compact groups in general. The book “Topological Groups and Related Structures” by Arhangel’skii and Tkachenko contains many results about such groups. In the 1960s Morris initiated the study of the classes of topological groups he called Varieties of Topological Groups, and several others have contributed to their theory. There has also been much research on pseudocompact groups by Comfort and his collaborators.

In this Topical Collection on Topological Groups we seek to address all areas of topological group theory and related structures. Original articles reporting recent progress and survey articles are sought. Authors are encouraged to include interesting open questions. The deadline for submitting papers is 31 December 2019.

Prof. Dr. Sidney A. Morris
Collection Editor

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Keywords

  • topological groups
  • compact groups
  • profinite groups
  • locally compact groups
  • Lie groups
  • actions by Lie groups
  • pro-Lie groups
  • almost periodic
  • semitopological groups
  • paratopological groups
  • structure theory
  • characterizing subgroups
  • transformation groups
  • representations
  • free topological groups, free Boolean topological groups and free products
  • variety of topological groups
  • Hilbert’s 5th problem
  • (locally) minimal topological groups
  • compactness conditions in topological groups
  • pseudocompact groups
  • duality and reflexivity
  • covering theory for topological groups
  • suitable sets for topological groups
  • algebraic topology and topological groups
  • compact semigroups

Related Special Issues

Published Papers (38 papers)

2022

Jump to: 2021, 2020, 2019, 2018, 2017, 2016, 2015

11 pages, 300 KiB  
Article
Topological and Geometrical Properties of k-Symplectic Structures
by Essabab Said, Fanich El Mokhtar and Awane Azzouz
Axioms 2022, 11(6), 273; https://doi.org/10.3390/axioms11060273 - 06 Jun 2022
Viewed by 1468
Abstract
We study new geometrical and topological aspects of polarized k-symplectic manifolds. In addition, we study the De Rham cohomology groups of the k-symplectic group. In this work, we pay particular attention to the problem of the orientation of polarized k-symplectic [...] Read more.
We study new geometrical and topological aspects of polarized k-symplectic manifolds. In addition, we study the De Rham cohomology groups of the k-symplectic group. In this work, we pay particular attention to the problem of the orientation of polarized k-symplectic manifolds in a way analogous to symplectic manifolds which are all orientable. Full article
5 pages, 234 KiB  
Article
Topological Transcendental Fields
by Taboka Prince Chalebgwa and Sidney A. Morris
Axioms 2022, 11(3), 118; https://doi.org/10.3390/axioms11030118 - 07 Mar 2022
Cited by 5 | Viewed by 1823
Abstract
This article initiates the study of topological transcendental fields F which are subfields of the topological field C of all complex numbers such that F only consists of rational numbers and a nonempty set of transcendental numbers. F, with the topology it [...] Read more.
This article initiates the study of topological transcendental fields F which are subfields of the topological field C of all complex numbers such that F only consists of rational numbers and a nonempty set of transcendental numbers. F, with the topology it inherits as a subspace of C, is a topological field. Each topological transcendental field is a separable metrizable zero-dimensional space and algebraically is Q(T), the extension of the field of rational numbers by a set T of transcendental numbers. It is proven that there exist precisely 20 countably infinite topological transcendental fields and each is homeomorphic to the space Q of rational numbers with its usual topology. It is also shown that there is a class of 220 of topological transcendental fields of the form Q(T) with T a set of Liouville numbers, no two of which are homeomorphic. Full article

2021

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5 pages, 216 KiB  
Article
Strictly Convex Banach Algebras
by David Yost
Axioms 2021, 10(3), 221; https://doi.org/10.3390/axioms10030221 - 11 Sep 2021
Viewed by 1876
Abstract
We discuss two facets of the interaction between geometry and algebra in Banach algebras. In the class of unital Banach algebras, there is essentially one known example which is also strictly convex as a Banach space. We recall this example, which is finite-dimensional, [...] Read more.
We discuss two facets of the interaction between geometry and algebra in Banach algebras. In the class of unital Banach algebras, there is essentially one known example which is also strictly convex as a Banach space. We recall this example, which is finite-dimensional, and consider the open question of generalising it to infinite dimensions. In C-algebras, we exhibit one striking example of the tighter relationship that exists between algebra and geometry there. Full article
12 pages, 316 KiB  
Article
Long Colimits of Topological Groups III: Homeomorphisms of Products and Coproducts
by Rafael Dahmen and Gábor Lukács
Axioms 2021, 10(3), 155; https://doi.org/10.3390/axioms10030155 - 19 Jul 2021
Viewed by 1334
Abstract
The group of compactly supported homeomorphisms on a Tychonoff space can be topologized in a number of ways, including as a colimit of homeomorphism groups with a given compact support or as a subgroup of the homeomorphism group of its Stone-Čech compactification. A [...] Read more.
The group of compactly supported homeomorphisms on a Tychonoff space can be topologized in a number of ways, including as a colimit of homeomorphism groups with a given compact support or as a subgroup of the homeomorphism group of its Stone-Čech compactification. A space is said to have the Compactly Supported Homeomorphism Property (CSHP) if these two topologies coincide. The authors provide necessary and sufficient conditions for finite products of ordinals equipped with the order topology to have CSHP. In addition, necessary conditions are presented for finite products and coproducts of spaces to have CSHP. Full article
6 pages, 234 KiB  
Article
A Semigroup Is Finite Iff It Is Chain-Finite and Antichain-Finite
by Iryna Banakh, Taras Banakh and Serhii Bardyla
Axioms 2021, 10(1), 9; https://doi.org/10.3390/axioms10010009 - 16 Jan 2021
Cited by 1 | Viewed by 1489
Abstract
A subset A of a semigroup S is called a chain (antichain) if ab{a,b} (ab{a,b}) for any (distinct) elements a,bA. [...] Read more.
A subset A of a semigroup S is called a chain (antichain) if ab{a,b} (ab{a,b}) for any (distinct) elements a,bA. A semigroup S is called periodic if for every element xS there exists nN such that xn is an idempotent. A semigroup S is called (anti)chain-finite if S contains no infinite (anti)chains. We prove that each antichain-finite semigroup S is periodic and for every idempotent e of S the set e={xS:nN(xn=e)} is finite. This property of antichain-finite semigroups is used to prove that a semigroup is finite if and only if it is chain-finite and antichain-finite. Furthermore, we present an example of an antichain-finite semilattice that is not a union of finitely many chains. Full article

2020

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9 pages, 246 KiB  
Article
Classes of Entire Analytic Functions of Unbounded Type on Banach Spaces
by Andriy Zagorodnyuk and Anna Hihliuk
Axioms 2020, 9(4), 133; https://doi.org/10.3390/axioms9040133 - 18 Nov 2020
Cited by 10 | Viewed by 1874
Abstract
In this paper we investigate analytic functions of unbounded type on a complex infinite dimensional Banach space X. The main question is: under which conditions is there an analytic function of unbounded type on X such that its Taylor polynomials are in [...] Read more.
In this paper we investigate analytic functions of unbounded type on a complex infinite dimensional Banach space X. The main question is: under which conditions is there an analytic function of unbounded type on X such that its Taylor polynomials are in prescribed subspaces of polynomials? We obtain some sufficient conditions for a function f to be of unbounded type and show that there are various subalgebras of polynomials that support analytic functions of unbounded type. In particular, some examples of symmetric analytic functions of unbounded type are constructed. Full article
9 pages, 226 KiB  
Article
Commutative Topological Semigroups Embedded into Topological Abelian Groups
by Julio César Hernández Arzusa
Axioms 2020, 9(3), 87; https://doi.org/10.3390/axioms9030087 - 24 Jul 2020
Cited by 3 | Viewed by 2383
Abstract
In this paper, we give conditions under which a commutative topological semigroup can be embedded algebraically and topologically into a compact topological Abelian group. We prove that every feebly compact regular first countable cancellative commutative topological semigroup with open shifts is a topological [...] Read more.
In this paper, we give conditions under which a commutative topological semigroup can be embedded algebraically and topologically into a compact topological Abelian group. We prove that every feebly compact regular first countable cancellative commutative topological semigroup with open shifts is a topological group, as well as every connected locally compact Hausdorff cancellative commutative topological monoid with open shifts. Finally, we use these results to give sufficient conditions on a commutative topological semigroup that guarantee it to have countable cellularity. Full article
11 pages, 283 KiB  
Article
Structure and Functions of Topological Metagroups
by Sergey V. Ludkowski
Axioms 2020, 9(2), 66; https://doi.org/10.3390/axioms9020066 - 14 Jun 2020
Cited by 3 | Viewed by 1875
Abstract
In this article, the structure of topological metagroups was investigated. Relations between topological and algebraic properties of metagroups were scrutinized. A uniform continuity of functions on them was studied. Smashed products of topological metagroups were investigated. Full article
7 pages, 265 KiB  
Article
On Grothendieck Sets
by Juan Carlos Ferrando, Salvador López-Alfonso and Manuel López-Pellicer
Axioms 2020, 9(1), 34; https://doi.org/10.3390/axioms9010034 - 24 Mar 2020
Cited by 5 | Viewed by 2319
Abstract
We call a subset M of an algebra of sets A a Grothendieck set for the Banach space b a ( A ) of bounded finitely additive scalar-valued measures on A equipped with the variation norm if each sequence [...] Read more.
We call a subset M of an algebra of sets A a Grothendieck set for the Banach space b a ( A ) of bounded finitely additive scalar-valued measures on A equipped with the variation norm if each sequence μ n n = 1 in b a ( A ) which is pointwise convergent on M is weakly convergent in b a ( A ) , i.e., if there is μ b a A such that μ n A μ A for every A M then μ n μ weakly in b a ( A ) . A subset M of an algebra of sets A is called a Nikodým set for b a ( A ) if each sequence μ n n = 1 in b a ( A ) which is pointwise bounded on M is bounded in b a ( A ) . We prove that if Σ is a σ -algebra of subsets of a set Ω which is covered by an increasing sequence Σ n : n N of subsets of Σ there exists p N such that Σ p is a Grothendieck set for b a ( A ) . This statement is the exact counterpart for Grothendieck sets of a classic result of Valdivia asserting that if a σ -algebra Σ is covered by an increasing sequence Σ n : n N of subsets, there is p N such that Σ p is a Nikodým set for b a Σ . This also refines the Grothendieck result stating that for each σ -algebra Σ the Banach space Σ is a Grothendieck space. Some applications to classic Banach space theory are given. Full article
18 pages, 369 KiB  
Article
Continuous Homomorphisms Defined on (Dense) Submonoids of Products of Topological Monoids
by Mikhail Tkachenko
Axioms 2020, 9(1), 23; https://doi.org/10.3390/axioms9010023 - 18 Feb 2020
Cited by 2 | Viewed by 2032
Abstract
We study the factorization properties of continuous homomorphisms defined on a (dense) submonoid S of a Tychonoff product D = i I D i of topological or even topologized monoids. In a number of different situations, we establish that every continuous [...] Read more.
We study the factorization properties of continuous homomorphisms defined on a (dense) submonoid S of a Tychonoff product D = i I D i of topological or even topologized monoids. In a number of different situations, we establish that every continuous homomorphism f : S K to a topological monoid (or group) K depends on at most finitely many coordinates. For example, this is the case if S is a subgroup of D and K is a first countable left topological group without small subgroups (i.e., K is an NSS group). A stronger conclusion is valid if S is a finitely retractable submonoid of D and K is a regular quasitopological NSS group of a countable pseudocharacter. In this case, every continuous homomorphism f of S to K has a finite type, which means that f admits a continuous factorization through a finite subproduct of D. A similar conclusion is obtained for continuous homomorphisms of submonoids (or subgroups) of products of topological monoids to Lie groups. Furthermore, we formulate a number of open problems intended to delimit the validity of our results. Full article
4 pages, 208 KiB  
Article
The Tubby Torus as a Quotient Group
by Sidney A. Morris
Axioms 2020, 9(1), 11; https://doi.org/10.3390/axioms9010011 - 20 Jan 2020
Cited by 1 | Viewed by 2159
Abstract
Let E be any metrizable nuclear locally convex space and E ^ the Pontryagin dual group of E. Then the topological group E ^ has the tubby torus (that is, the countably infinite product of copies of the circle group) as a [...] Read more.
Let E be any metrizable nuclear locally convex space and E ^ the Pontryagin dual group of E. Then the topological group E ^ has the tubby torus (that is, the countably infinite product of copies of the circle group) as a quotient group if and only if E does not have the weak topology. This extends results in the literature related to the Banach–Mazur separable quotient problem. Full article
5 pages, 213 KiB  
Article
Observations on the Separable Quotient Problem for Banach Spaces
by Sidney A. Morris and David T. Yost
Axioms 2020, 9(1), 7; https://doi.org/10.3390/axioms9010007 - 13 Jan 2020
Cited by 2 | Viewed by 2329
Abstract
The longstanding Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space has a quotient (Banach) space that is both infinite-dimensional and separable. Although it remains open in general, an affirmative answer is known in many special cases, including (1) reflexive Banach spaces, [...] Read more.
The longstanding Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space has a quotient (Banach) space that is both infinite-dimensional and separable. Although it remains open in general, an affirmative answer is known in many special cases, including (1) reflexive Banach spaces, (2) weakly compactly generated (WCG) spaces, and (3) Banach spaces which are dual spaces. Obviously (1) is a special case of both (2) and (3), but neither (2) nor (3) is a special case of the other. A more general result proved here includes all three of these cases. More precisely, we call an infinite-dimensional Banach space X dual-like, if there is another Banach space E, a continuous linear operator T from the dual space E * onto a dense subspace of X, such that the closure of the kernel of T (in the relative weak* topology) has infinite codimension in E * . It is shown that every dual-like Banach space has an infinite-dimensional separable quotient. Full article

2019

Jump to: 2022, 2021, 2020, 2018, 2017, 2016, 2015

10 pages, 295 KiB  
Review
A Short Survey and Open Questions on Compact Q-Groups
by Francesco G. Russo
Axioms 2019, 8(4), 128; https://doi.org/10.3390/axioms8040128 - 13 Nov 2019
Viewed by 2094
Abstract
Finite Q -groups have been recently studied and form a class of solvable groups, which satisfy interesting structural conditions. We survey some of their main properties and introduce the idea of Q -group for compact p-groups (p prime). A list of [...] Read more.
Finite Q -groups have been recently studied and form a class of solvable groups, which satisfy interesting structural conditions. We survey some of their main properties and introduce the idea of Q -group for compact p-groups (p prime). A list of open questions is presented, along with several connections of arithmetic nature on a problem originally due to Frobenius. Full article
16 pages, 400 KiB  
Article
Structure of Finite-Dimensional Protori
by Wayne Lewis
Axioms 2019, 8(3), 93; https://doi.org/10.3390/axioms8030093 - 01 Aug 2019
Viewed by 3360
Abstract
A Structure Theorem for Protori is derived for the category of finite-dimensional protori (compact connected abelian groups), which details the interplay between the properties of density, discreteness, torsion, and divisibility within a finite-dimensional protorus. The spectrum of resolutions for a finite-dimensional protorus are [...] Read more.
A Structure Theorem for Protori is derived for the category of finite-dimensional protori (compact connected abelian groups), which details the interplay between the properties of density, discreteness, torsion, and divisibility within a finite-dimensional protorus. The spectrum of resolutions for a finite-dimensional protorus are parameterized in the structure theorem by the dual category of finite rank torsion-free abelian groups. A consequence is a universal resolution for a finite-dimensional protorus, independent of a choice of a particular subgroup. A resolution is also given strictly in terms of the path component of the identity and the union of all zero-dimensional subgroups. The structure theorem is applied to show that a morphism of finite-dimensional protori lifts to a product morphism between products of periodic locally compact groups and real vector spaces. Full article
22 pages, 382 KiB  
Article
Eilenberg–Mac Lane Spaces for Topological Groups
by Ged Corob Cook
Axioms 2019, 8(3), 90; https://doi.org/10.3390/axioms8030090 - 27 Jul 2019
Viewed by 2861
Abstract
In this paper, we establish a topological version of the notion of an Eilenberg–Mac Lane space. If X is a pointed topological space, π 1 ( X ) has a natural topology coming from the compact-open topology on the space of maps [...] Read more.
In this paper, we establish a topological version of the notion of an Eilenberg–Mac Lane space. If X is a pointed topological space, π 1 ( X ) has a natural topology coming from the compact-open topology on the space of maps S 1 X . In general, the construction does not produce a topological group because it is possible to create examples where the group multiplication π 1 ( X ) × π 1 ( X ) π 1 ( X ) is discontinuous. This discontinuity has been noticed by others, for example Fabel. However, if we work in the category of compactly generated, weakly Hausdorff spaces, we may retopologise both the space of maps S 1 X and the product π 1 ( X ) × π 1 ( X ) with compactly generated topologies to see that π 1 ( X ) is a group object in this category. Such group objects are known as k-groups. Next we construct the Eilenberg–Mac Lane space K ( G , 1 ) for any totally path-disconnected k-group G. The main point of this paper is to show that, for such a G, π 1 ( K ( G , 1 ) ) is isomorphic to G in the category of k-groups. All totally disconnected locally compact groups are k-groups and so our results apply in particular to profinite groups, answering a question of Sauer’s. We also show that analogues of the Mayer–Vietoris sequence and Seifert–van Kampen theorem hold in this context. The theory requires a careful analysis using model structures and other homotopical structures on cartesian closed categories as we shall see that no theory can be comfortably developed in the classical world. Full article
10 pages, 272 KiB  
Article
Factoring Continuous Homomorphisms Defined on Submonoids of Products of Topologized Monoids
by Mikhail Tkachenko
Axioms 2019, 8(3), 86; https://doi.org/10.3390/axioms8030086 - 26 Jul 2019
Cited by 4 | Viewed by 2502
Abstract
We study factorization properties of continuous homomorphisms defined on submonoids of products of topologized monoids. We prove that if S is an ω-retractable submonoid of a product D = i I D i of topologized monoids and [...] Read more.
We study factorization properties of continuous homomorphisms defined on submonoids of products of topologized monoids. We prove that if S is an ω-retractable submonoid of a product D = i I D i of topologized monoids and f : S H is a continuous homomorphism to a topologized semigroup H with ψ ( H ) ω , then one can find a countable subset E of I and a continuous homomorphism g : p E ( S ) H satisfying f = g p E S , where p E is the projection of D to i E D i . The same conclusion is valid if S contains the Σ -product Σ D D . Furthermore, we show that in both cases, there exists the smallest by inclusion subset E I with the aforementioned properties. Full article
6 pages, 242 KiB  
Article
Hereditary Coreflective Subcategories in Certain Categories of Abelian Semitopological Groups
by Veronika Pitrová
Axioms 2019, 8(3), 85; https://doi.org/10.3390/axioms8030085 - 24 Jul 2019
Viewed by 2718
Abstract
Let A be an epireflective subcategory of the category of all semitopological groups that consists only of abelian groups. We describe maximal hereditary coreflective subcategories of A that are not bicoreflective in A in the case that the A -reflection of the discrete [...] Read more.
Let A be an epireflective subcategory of the category of all semitopological groups that consists only of abelian groups. We describe maximal hereditary coreflective subcategories of A that are not bicoreflective in A in the case that the A -reflection of the discrete group of integers is a finite cyclic group, the group of integers with a topology that is not T 0 , or the group of integers with the topology generated by its subgroups of the form p n , where n N , p P and P is a given set of prime numbers. Full article

2018

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18 pages, 345 KiB  
Article
Separability of Topological Groups: A Survey with Open Problems
by Arkady G. Leiderman and Sidney A. Morris
Axioms 2019, 8(1), 3; https://doi.org/10.3390/axioms8010003 - 29 Dec 2018
Cited by 6 | Viewed by 3779
Abstract
Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C ( K ) for metrizable compact spaces K [...] Read more.
Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C ( K ) for metrizable compact spaces K; and p , for p 1 . This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey. Full article
5 pages, 220 KiB  
Article
(L)-Semigroup Sums
by John R. Martin
Axioms 2019, 8(1), 1; https://doi.org/10.3390/axioms8010001 - 22 Dec 2018
Cited by 1 | Viewed by 2688
Abstract
An (L)-semigroup S is a compact n-manifold with connected boundary B together with a monoid structure on S such that B is a subsemigroup of S. The sum S + T of two (L)-semigroups S and T having boundary B is [...] Read more.
An (L)-semigroup S is a compact n-manifold with connected boundary B together with a monoid structure on S such that B is a subsemigroup of S. The sum S + T of two (L)-semigroups S and T having boundary B is the quotient space obtained from the union of S × { 0 } and T × { 1 } by identifying the point ( x , 0 ) in S × { 0 } with ( x , 1 ) in T × { 1 } for each x in B. It is shown that no (L)-semigroup sum of dimension less than or equal to five admits an H-space structure, nor does any (L)-semigroup sum obtained from (L)-semigroups having an Abelian boundary. In particular, such sums cannot be a retract of a topological group. Full article
23 pages, 416 KiB  
Article
Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets
by Dmitri Shakhmatov and Víctor Hugo Yañez
Axioms 2018, 7(4), 86; https://doi.org/10.3390/axioms7040086 - 16 Nov 2018
Cited by 3 | Viewed by 3131
Abstract
We give a “naive” (i.e., using no additional set-theoretic assumptions beyond ZFC, the Zermelo-Fraenkel axioms of set theory augmented by the Axiom of Choice) example of a Boolean topological group G without infinite separable pseudocompact subsets having the following “selective” compactness property: For [...] Read more.
We give a “naive” (i.e., using no additional set-theoretic assumptions beyond ZFC, the Zermelo-Fraenkel axioms of set theory augmented by the Axiom of Choice) example of a Boolean topological group G without infinite separable pseudocompact subsets having the following “selective” compactness property: For each free ultrafilter p on the set N of natural numbers and every sequence ( U n ) of non-empty open subsets of G, one can choose a point x n U n for all n N in such a way that the resulting sequence ( x n ) has a p-limit in G; that is, { n N : x n V } p for every neighbourhood V of x in G. In particular, G is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. This answers a question of Dorantes-Aldama and the first listed author. The group G above is not pseudo- ω -bounded either. Furthermore, we show that the free precompact Boolean group of a topological sum i I X i , where each space X i is either maximal or discrete, contains no infinite separable pseudocompact subsets. Full article
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4 pages, 207 KiB  
Article
Extending Characters of Fixed Point Algebras
by Stefan Wagner
Axioms 2018, 7(4), 79; https://doi.org/10.3390/axioms7040079 - 07 Nov 2018
Cited by 2 | Viewed by 2414
Abstract
A dynamical system is a triple ( A , G , α ) consisting of a unital locally convex algebra A, a topological group G, and a group homomorphism α : G Aut ( A ) that induces a continuous [...] Read more.
A dynamical system is a triple ( A , G , α ) consisting of a unital locally convex algebra A, a topological group G, and a group homomorphism α : G Aut ( A ) that induces a continuous action of G on A. Furthermore, a unital locally convex algebra A is called a continuous inverse algebra, or CIA for short, if its group of units A × is open in A and the inversion map ι : A × A × , a a 1 is continuous at 1 A . Given a dynamical system ( A , G , α ) with a complete commutative CIA A and a compact group G, we show that each character of the corresponding fixed point algebra can be extended to a character of A. Full article
6 pages, 242 KiB  
Article
A Note on the Topological Group c0
by Michael Megrelishvili
Axioms 2018, 7(4), 77; https://doi.org/10.3390/axioms7040077 - 29 Oct 2018
Cited by 1 | Viewed by 2449
Abstract
A well-known result of Ferri and Galindo asserts that the topological group c 0 is not reflexively representable and the algebra WAP ( c 0 ) of weakly almost periodic functions does not separate points and closed subsets. However, it is unknown if [...] Read more.
A well-known result of Ferri and Galindo asserts that the topological group c 0 is not reflexively representable and the algebra WAP ( c 0 ) of weakly almost periodic functions does not separate points and closed subsets. However, it is unknown if the same remains true for a larger important algebra Tame ( c 0 ) of tame functions. Respectively, it is an open question if c 0 is representable on a Rosenthal Banach space. In the present work we show that Tame ( c 0 ) is small in a sense that the unit sphere S and 2 S cannot be separated by a tame function f ∈ Tame ( c 0 ) . As an application we show that the Gromov’s compactification of c 0 is not a semigroup compactification. We discuss some questions. Full article
12 pages, 271 KiB  
Review
Selective Survey on Spaces of Closed Subgroups of Topological Groups
by Igor V. Protasov
Axioms 2018, 7(4), 75; https://doi.org/10.3390/axioms7040075 - 26 Oct 2018
Viewed by 2528
Abstract
We survey different topologizations of the set S ( G ) of closed subgroups of a topological group G and demonstrate some applications using Topological Groups, Model Theory, Geometric Group Theory, and Topological Dynamics. [...] Read more.
We survey different topologizations of the set S ( G ) of closed subgroups of a topological group G and demonstrate some applications using Topological Groups, Model Theory, Geometric Group Theory, and Topological Dynamics. Full article
5 pages, 239 KiB  
Article
Varieties of Coarse Spaces
by Igor Protasov
Axioms 2018, 7(2), 32; https://doi.org/10.3390/axioms7020032 - 14 May 2018
Cited by 3 | Viewed by 3092
Abstract
A class M of coarse spaces is called a variety if M is closed under the formation of subspaces, coarse images, and products. We classify the varieties of coarse spaces and, in particular, show that if a variety M contains an unbounded metric [...] Read more.
A class M of coarse spaces is called a variety if M is closed under the formation of subspaces, coarse images, and products. We classify the varieties of coarse spaces and, in particular, show that if a variety M contains an unbounded metric space then M is the variety of all coarse spaces. Full article

2017

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324 KiB  
Article
Large Sets in Boolean and Non-Boolean Groups and Topology
by Ol’ga V. Sipacheva
Axioms 2017, 6(4), 28; https://doi.org/10.3390/axioms6040028 - 24 Oct 2017
Viewed by 3456
Abstract
Various notions of large sets in groups, including the classical notions of thick, syndetic, and piecewise syndetic sets and the new notion of vast sets in groups, are studied with emphasis on the interplay between such sets in Boolean groups. Natural topologies closely [...] Read more.
Various notions of large sets in groups, including the classical notions of thick, syndetic, and piecewise syndetic sets and the new notion of vast sets in groups, are studied with emphasis on the interplay between such sets in Boolean groups. Natural topologies closely related to vast sets are considered; as a byproduct, interesting relations between vast sets and ultrafilters are revealed. Full article
299 KiB  
Article
Computing the Scale of an Endomorphism of a totally Disconnected Locally Compact Group
by George A. Willis
Axioms 2017, 6(4), 27; https://doi.org/10.3390/axioms6040027 - 20 Oct 2017
Cited by 1 | Viewed by 4588
Abstract
The scale of an endomorphism of a totally disconnected, locally compact group G is defined and an example is presented which shows that the scale function is not always continuous with respect to the Braconnier topology on the automorphism group of G. [...] Read more.
The scale of an endomorphism of a totally disconnected, locally compact group G is defined and an example is presented which shows that the scale function is not always continuous with respect to the Braconnier topology on the automorphism group of G. Methods for computing the scale, which is a positive integer, are surveyed and illustrated by applying them in diverse cases, including when G is compact; an automorphism group of a tree; Neretin’s group of almost automorphisms of a tree; and a p-adic Lie group. The information required to compute the scale is reviewed from the perspective of the, as yet incomplete, general theory of totally disconnected, locally compact groups. Full article
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433 KiB  
Article
Categorically Closed Topological Groups
by Taras Banakh
Axioms 2017, 6(3), 23; https://doi.org/10.3390/axioms6030023 - 30 Jul 2017
Cited by 11 | Viewed by 4591
Abstract
Let \({\overset{\rightarrow}{\mathcal{C}} }\) be a category whose objects are semigroups with topology and morphisms are closed semigroup relations, in particular, continuous homomorphisms. An object X of the category \({\overset{\rightarrow}{\mathcal{C}} }\) is called \({\overset{\rightarrow}{\mathcal{C}} }\)-closed if for each morphism \({\Phi\subset X\times Y}\) in [...] Read more.
Let \({\overset{\rightarrow}{\mathcal{C}} }\) be a category whose objects are semigroups with topology and morphisms are closed semigroup relations, in particular, continuous homomorphisms. An object X of the category \({\overset{\rightarrow}{\mathcal{C}} }\) is called \({\overset{\rightarrow}{\mathcal{C}} }\)-closed if for each morphism \({\Phi\subset X\times Y}\) in the category \({\overset{\rightarrow}{\mathcal{C}} }\) the image \({\Phi(X)=\{y\in Y:\exists x\in X\;(x,y)\in\Phi\}}\) is closed in Y. In the paper we survey existing and new results on topological groups, which are \({\overset{\rightarrow}{\mathcal{C}} }\)-closed for various categories \({\overset{\rightarrow}{\mathcal{C}} }\) of topologized semigroups. Full article
220 KiB  
Article
No Uncountable Polish Group Can be a Right-Angled Artin Group
by Gianluca Paolini and Saharon Shelah
Axioms 2017, 6(2), 13; https://doi.org/10.3390/axioms6020013 - 11 May 2017
Cited by 3 | Viewed by 3412
Abstract
We prove that if G is a Polish group and A a group admitting a system of generators whose associated length function satisfies: (i) if 0 < k < ω , then [...] Read more.
We prove that if G is a Polish group and A a group admitting a system of generators whose associated length function satisfies: (i) if 0 < k < ω , then l g ( x ) l g ( x k ) ; (ii) if l g ( y ) < k < ω and x k = y , then x = e , then there exists a subgroup G * of G of size b (the bounding number) such that G * is not embeddable in A. In particular, we prove that the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes analogous results for free and free abelian uncountable groups. Full article

2016

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220 KiB  
Editorial
An Overview of Topological Groups: Yesterday, Today, Tomorrow
by Sidney A. Morris
Axioms 2016, 5(2), 11; https://doi.org/10.3390/axioms5020011 - 05 May 2016
Viewed by 4365
Abstract
It was in 1969 that I began my graduate studies on topological group theory and I often dived into one of the following five books. My favourite book “Abstract Harmonic Analysis” [1] by Ed Hewitt and Ken Ross contains both a proof of [...] Read more.
It was in 1969 that I began my graduate studies on topological group theory and I often dived into one of the following five books. My favourite book “Abstract Harmonic Analysis” [1] by Ed Hewitt and Ken Ross contains both a proof of the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups and the structure theory of locally compact abelian groups.[...] Full article
296 KiB  
Article
Non-Abelian Pseudocompact Groups
by W. W. Comfort and Dieter Remus
Axioms 2016, 5(1), 2; https://doi.org/10.3390/axioms5010002 - 12 Jan 2016
Cited by 3 | Viewed by 4107
Abstract
Here are three recently-established theorems from the literature. (A) (2006) Every non-metrizable compact abelian group K has 2|K| -many proper dense pseudocompact subgroups. (B) (2003) Every non-metrizable compact abelian group K admits 22|K| -many strictly finer pseudocompact topological group refinements. [...] Read more.
Here are three recently-established theorems from the literature. (A) (2006) Every non-metrizable compact abelian group K has 2|K| -many proper dense pseudocompact subgroups. (B) (2003) Every non-metrizable compact abelian group K admits 22|K| -many strictly finer pseudocompact topological group refinements. (C) (2007) Every non-metrizable pseudocompact abelian group has a proper dense pseudocompact subgroup and a strictly finer pseudocompact topological group refinement. (Theorems (A), (B) and (C) become false if the non-metrizable hypothesis is omitted.) With a detailed view toward the relevant literature, the present authors ask: What happens to (A), (B), (C) and to similar known facts about pseudocompact abelian groups if the abelian hypothesis is omitted? Are the resulting statements true, false, true under certain natural additional hypotheses, etc.? Several new results responding in part to these questions are given, and several specific additional questions are posed. Full article

2015

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300 KiB  
Article
Free Boolean Topological Groups
by Ol’ga Sipacheva
Axioms 2015, 4(4), 492-517; https://doi.org/10.3390/axioms4040492 - 03 Nov 2015
Cited by 12 | Viewed by 4921
Abstract
Known and new results on free Boolean topological groups are collected. An account of the properties that these groups share with free or free Abelian topological groups and properties specific to free Boolean groups is given. Special emphasis is placed on the application [...] Read more.
Known and new results on free Boolean topological groups are collected. An account of the properties that these groups share with free or free Abelian topological groups and properties specific to free Boolean groups is given. Special emphasis is placed on the application of set-theoretic methods to the study of Boolean topological groups. Full article
352 KiB  
Article
Characterized Subgroups of Topological Abelian Groups
by Dikran Dikranjan, Anna Giordano Bruno and Daniele Impieri
Axioms 2015, 4(4), 459-491; https://doi.org/10.3390/axioms4040459 - 16 Oct 2015
Cited by 9 | Viewed by 4492
Abstract
A subgroup H of a topological abelian group X is said to be characterized by a sequence v = (vn) of characters of X if H = {xX : vn(x) → 0 in T}. [...] Read more.
A subgroup H of a topological abelian group X is said to be characterized by a sequence v = (vn) of characters of X if H = {xX : vn(x) → 0 in T}. We study the basic properties of characterized subgroups in the general setting, extending results known in the compact case. For a better description, we isolate various types of characterized subgroups. Moreover, we introduce the relevant class of auto-characterized groups (namely, the groups that are characterized subgroups of themselves by means of a sequence of non-null characters); in the case of locally compact abelian groups, these are proven to be exactly the non-compact ones. As a by-product of our results, we find a complete description of the characterized subgroups of discrete abelian groups. Full article
336 KiB  
Article
Locally Quasi-Convex Compatible Topologies on a Topological Group
by Lydia Außenhofer, Dikran Dikranjan and Elena Martín-Peinador
Axioms 2015, 4(4), 436-458; https://doi.org/10.3390/axioms4040436 - 13 Oct 2015
Cited by 9 | Viewed by 4737
Abstract
For a locally quasi-convex topological abelian group (G,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (G) that are compatible with (τ) (i.e., have the same dual as (G,τ) ordered by inclusion. [...] Read more.
For a locally quasi-convex topological abelian group (G,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (G) that are compatible with (τ) (i.e., have the same dual as (G,τ) ordered by inclusion. Obviously, this poset has always a bottom element, namely the weak topology σ(G,\(\widehat{G})\) . Whether it has also a top element is an open question. We study both quantitative aspects of this poset (its size) and its qualitative aspects, e.g., its chains and anti-chains. Since we are mostly interested in estimates ``from below'', our strategy consists of finding appropriate subgroups (H) of (G) that are easier to handle and show that \(\mathscr{C} (H)\) and \(\mathscr{C} (G/H)\) are large and embed, as a poset, in \(\mathscr{C}(G,τ)\). Important special results are: (i) if \(K\) is a compact subgroup of a locally quasi-convex group \(G\), then \(\mathscr{C}(G)\) and \(\mathscr{C}(G/K)\) are quasi-isomorphic (3.15); (ii) if (D) is a discrete abelian group of infinite rank, then \(\mathscr{C}(D)\) is quasi-isomorphic to the poset \(\mathfrak{F}_D\) of filters on D (4.5). Combining both results, we prove that for an LCA (locally compact abelian) group \(G \) with an open subgroup of infinite co-rank (this class includes, among others, all non-σ-compact LCA groups), the poset \( \mathscr{C} (G) \) is as big as the underlying topological structure of (G,τ) (and set theory) allows. For a metrizable connected compact group \(X\), the group of null sequences \(G=c_0(X)\) with the topology of uniform convergence is studied. We prove that \(\mathscr{C}(G)\) is quasi-isomorphic to \(\mathscr{P}(\mathbb{R})\) (6.9). Full article
192 KiB  
Article
Fixed Points of Local Actions of Lie Groups on Real and Complex 2-Manifolds
by Morris W. Hirsch
Axioms 2015, 4(3), 313-320; https://doi.org/10.3390/axioms4030313 - 27 Jul 2015
Cited by 2 | Viewed by 4240
Abstract
I discuss old and new results on fixed points of local actions by Lie groups G on real and complex 2-manifolds, and zero sets of Lie algebras of vector fields. Results of E. Lima, J. Plante and C. Bonatti are reviewed. Full article
263 KiB  
Article
Pro-Lie Groups: A Survey with Open Problems
by Karl H. Hofmann and Sidney A. Morris
Axioms 2015, 4(3), 294-312; https://doi.org/10.3390/axioms4030294 - 24 Jul 2015
Cited by 15 | Viewed by 5562
Abstract
A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete [...] Read more.
A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally-compact group that has a compact quotient group modulo its identity component and, thus, in particular, each compact and each connected locally-compact group; it also includes all locally-compact Abelian groups. This paper provides an overview of the structure theory and the Lie theory of pro-Lie groups, including results more recent than those in the authors’ reference book on pro-Lie groups. Significantly, it also includes a review of the recent insight that weakly-complete unital algebras provide a natural habitat for both pro-Lie algebras and pro-Lie groups, indeed for the exponential function that links the two. (A topological vector space is weakly complete if it is isomorphic to a power RX of an arbitrary set of copies of R. This class of real vector spaces is at the basis of the Lie theory of pro-Lie groups.) The article also lists 12 open questions connected to pro-Lie groups. Full article
235 KiB  
Article
Lindelöf Σ-Spaces and R-Factorizable Paratopological Groups
by Mikhail Tkachenko
Axioms 2015, 4(3), 254-267; https://doi.org/10.3390/axioms4030254 - 10 Jul 2015
Cited by 2 | Viewed by 4196
Abstract
We prove that if a paratopological group G is a continuous image of an arbitrary product of regular Lindelöf Σ-spaces, then it is R-factorizable and has countable cellularity. If in addition, G is regular, then it is totally w-narrow and satisfies celw [...] Read more.
We prove that if a paratopological group G is a continuous image of an arbitrary product of regular Lindelöf Σ-spaces, then it is R-factorizable and has countable cellularity. If in addition, G is regular, then it is totally w-narrow and satisfies celw(G) ≤ w, and the Hewitt–Nachbin completion of G is again an R-factorizable paratopological group. Full article
283 KiB  
Article
On T-Characterized Subgroups of Compact Abelian Groups
by Saak Gabriyelyan
Axioms 2015, 4(2), 194-212; https://doi.org/10.3390/axioms4020194 - 19 Jun 2015
Cited by 4 | Viewed by 4454
Abstract
A sequence \(\{ u_n \}_{n\in \omega}\) in abstract additively-written Abelian group \(G\) is called a \(T\)-sequence if there is a Hausdorff group topology on \(G\) relative to which \(\lim_n u_n =0\). We say that a subgroup \(H\) of an infinite compact Abelian group [...] Read more.
A sequence \(\{ u_n \}_{n\in \omega}\) in abstract additively-written Abelian group \(G\) is called a \(T\)-sequence if there is a Hausdorff group topology on \(G\) relative to which \(\lim_n u_n =0\). We say that a subgroup \(H\) of an infinite compact Abelian group \(X\) is \(T\)-characterized if there is a \(T\)-sequence \(\mathbf{u} =\{ u_n \}\) in the dual group of \(X\), such that \(H=\{ x\in X: \; (u_n, x)\to 1 \}\). We show that a closed subgroup \(H\) of \(X\) is \(T\)-characterized if and only if \(H\) is a \(G_\delta\)-subgroup of \(X\) and the annihilator of \(H\) admits a Hausdorff minimally almost periodic group topology. All closed subgroups of an infinite compact Abelian group \(X\) are \(T\)-characterized if and only if \(X\) is metrizable and connected. We prove that every compact Abelian group \(X\) of infinite exponent has a \(T\)-characterized subgroup, which is not an \(F_{\sigma}\)-subgroup of \(X\), that gives a negative answer to Problem 3.3 in Dikranjan and Gabriyelyan (Topol. Appl. 2013, 160, 2427–2442). Full article
267 KiB  
Review
Open and Dense Topological Transitivity of Extensions by Non-Compact Fiber of Hyperbolic Systems: A Review
by Viorel Nitica and Andrei Török
Axioms 2015, 4(1), 84-101; https://doi.org/10.3390/axioms4010084 - 04 Feb 2015
Cited by 2 | Viewed by 4761
Abstract
Currently, there is great renewed interest in proving the topological transitivity of various classes of continuous dynamical systems. Even though this is one of the most basic dynamical properties that can be investigated, the tools used by various authors are quite diverse and [...] Read more.
Currently, there is great renewed interest in proving the topological transitivity of various classes of continuous dynamical systems. Even though this is one of the most basic dynamical properties that can be investigated, the tools used by various authors are quite diverse and are strongly related to the class of dynamical systems under consideration. The goal of this review article is to present the state of the art for the class of Hölder extensions of hyperbolic systems with non-compact connected Lie group fiber. The hyperbolic systems we consider are mostly discrete time. In particular, we address the stability and genericity of topological transitivity in large classes of such transformations. The paper lists several open problems and conjectures and tries to place this topic of research in the general context of hyperbolic and topological dynamics. Full article
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