**Abstract**

*K*[...] Read more.

*K*; and

A topical collection in *Axioms* (ISSN 2075-1680).

Collection Editor
Prof. Dr. Sidney A. Morris
1. Centre for Informatics and Applied Optimization, Federation University Australia, Victoria 3353, Australia
2. School of Engineering and Mathematical Sciences, La Trobe University, Bundoora, Victoria 3086, Australia Website | E-Mail Interests: topological groups especially locally compact groups; topology; group theory; functional analysis; universal algebra; 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 |

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 30 June 2019**.

Prof. Dr. Sidney A. Morris*Collection Editor*

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. *Axioms* is an international peer-reviewed open access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 350 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

- 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

- Topological Groups: Yesterday, Today, Tomorrow in
*Axioms*(10 articles - displayed below)

Separability of Topological Groups: A Survey with Open Problems
**Abstract **

Received: 25 November 2018 / Revised: 23 December 2018 / Accepted: 25 December 2018 / Published: 29 December 2018

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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 $C\left(K\right)$ for metrizable compact spaces *K*
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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 $C\left(K\right)$ for metrizable compact spaces *K*; and ${\ell}_{p}$ , for $p\ge 1$ . 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.
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(L)-Semigroup Sums
**Abstract **

Received: 12 November 2018 / Revised: 14 December 2018 / Accepted: 17 December 2018 / Published: 22 December 2018

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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
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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 $S\times \left\{0\right\}$ and $T\times \left\{1\right\}$ by identifying the point $(x,0)$ in $S\times \left\{0\right\}$ with $(x,1)$ in $T\times \left\{1\right\}$ 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

Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets
**Abstract **
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Received: 31 July 2018 / Revised: 21 October 2018 / Accepted: 5 November 2018 / Published: 16 November 2018

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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 $\mathbb{N}$ of natural numbers and every sequence $\left({U}_{n}\right)$ of non-empty open subsets of *G*, one can choose a point ${x}_{n}\in {U}_{n}$ for all $n\in \mathbb{N}$ in such a way that the resulting sequence $\left({x}_{n}\right)$ has a *p*-limit in *G*; that is, $\{n\in \mathbb{N}:{x}_{n}\in V\}\in p$ 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- $\omega $ -bounded either. Furthermore, we show that the free precompact Boolean group of a topological sum ${\u2a01}_{i\in I}{X}_{i}$ , where each space ${X}_{i}$ is either maximal or discrete, contains no infinite separable pseudocompact subsets.
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Extending Characters of Fixed Point Algebras
**Abstract **

Received: 13 October 2018 / Revised: 2 November 2018 / Accepted: 5 November 2018 / Published: 7 November 2018

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A dynamical system is a triple $(A,G,\alpha )$ consisting of a unital locally convex algebra *A*, a topological group *G*, and a group homomorphism $\alpha :G\to Aut\left(A\right)$ that induces a continuous
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A dynamical system is a triple $(A,G,\alpha )$ consisting of a unital locally convex algebra *A*, a topological group *G*, and a group homomorphism $\alpha :G\to Aut\left(A\right)$ 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 ${A}^{\times}$ is open in *A* and the inversion map $\iota :{A}^{\times}\to {A}^{\times}$ , $a\mapsto {a}^{-1}$ is continuous at ${1}_{A}$ . Given a dynamical system $(A,G,\alpha )$ 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*.
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A Note on the Topological Group *c*_{0}
**Abstract **

Received: 28 September 2018 / Revised: 22 October 2018 / Accepted: 24 October 2018 / Published: 29 October 2018

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A well-known result of Ferri and Galindo asserts that the topological group ${c}_{0}$ is not reflexively representable and the algebra WAP$\left({c}_{0}\right)$ 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 $\left({c}_{0}\right)$ 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 $\left({c}_{0}\right)$ 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 $\left({c}_{0}\right)$ is small in a sense that the unit sphere *S* and $2S$ cannot be separated by a tame function *f* ∈ Tame $\left({c}_{0}\right)$ . As an application we show that the Gromov’s compactification of ${c}_{0}$ is not a semigroup compactification. We discuss some questions.
Full article

Selective Survey on Spaces of Closed Subgroups of Topological Groups
**Abstract **

Received: 8 October 2018 / Revised: 24 October 2018 / Accepted: 24 October 2018 / Published: 26 October 2018

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We survey different topologizations of the set $\mathcal{S}\left(G\right)$ of closed subgroups of a topological group *G* and demonstrate some applications using Topological Groups, Model Theory, Geometric Group Theory, and Topological Dynamics.
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We survey different topologizations of the set $\mathcal{S}\left(G\right)$ of closed subgroups of a topological group *G* and demonstrate some applications using Topological Groups, Model Theory, Geometric Group Theory, and Topological Dynamics.
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Varieties of Coarse Spaces
**Abstract **

Received: 27 March 2018 / Revised: 22 April 2018 / Accepted: 10 May 2018 / Published: 14 May 2018

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A class $\mathfrak{M}$ of coarse spaces is called a variety if $\mathfrak{M}$ 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 $\mathfrak{M}$ contains an unbounded metric
[...] Read more.

A class $\mathfrak{M}$ of coarse spaces is called a variety if $\mathfrak{M}$ 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 $\mathfrak{M}$ contains an unbounded metric space then $\mathfrak{M}$ is the variety of all coarse spaces.
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Large Sets in Boolean and Non-Boolean Groups and Topology
**Abstract **

Received: 1 September 2017 / Revised: 20 October 2017 / Accepted: 23 October 2017 / Published: 24 October 2017

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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
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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.
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Computing the Scale of an Endomorphism of a totally Disconnected Locally Compact Group
**Abstract **
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Received: 28 August 2017 / Revised: 9 October 2017 / Accepted: 9 October 2017 / Published: 20 October 2017

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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*.
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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.
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Categorically Closed Topological Groups
**Abstract **

Received: 30 June 2017 / Revised: 26 July 2017 / Accepted: 27 July 2017 / Published: 30 July 2017

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Let 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 C → is called C → -closed if for each morphism Φ ⊂ X × Y in the category C → the image Φ ( X ) = { y ∈ Y : ∃ x ∈ X ( x , y ) ∈ Φ } is closed in Y . In the paper we survey existing and new results on topological groups, which are C → -closed for various categories C → of topologized semigroups.
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No Uncountable Polish Group Can be a Right-Angled Artin Group
**Abstract **

Received: 28 March 2017 / Revised: 20 April 2017 / Accepted: 4 May 2017 / Published: 11 May 2017

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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<\omega $ , then $lg\left(x\right)\le lg({x}^{k}$
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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<\omega $ , then $lg\left(x\right)\le lg\left({x}^{k}\right)$ ; (ii) if $lg\left(y\right)<k<\omega $ and ${x}^{k}=y$ , then $x=e$ , then there exists a subgroup ${G}^{*}$ of *G* of size $\mathfrak{b}$ (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.
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An Overview of Topological Groups: Yesterday, Today, Tomorrow
**Abstract **

Received: 18 April 2016 / Accepted: 20 April 2016 / Published: 5 May 2016

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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.[...]
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Non-Abelian Pseudocompact Groups
**Abstract **

Received: 29 September 2015 / Revised: 24 November 2015 / Accepted: 23 December 2015 / Published: 12 January 2016

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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 2^{2|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 2^{2|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.
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Free Boolean Topological Groups
**Abstract **

Received: 30 August 2015 / Revised: 17 October 2015 / Accepted: 23 October 2015 / Published: 3 November 2015

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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.
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Characterized Subgroups of Topological Abelian Groups
**Abstract **

Received: 2 September 2015 / Revised: 27 September 2015 / Accepted: 8 October 2015 / Published: 16 October 2015

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A subgroup H of a topological abelian group X is said to be characterized by a sequence **v** = (v_{n}) of characters of *X* if *H* = {*x* ∈ *X* : v_{n}(*x*) → 0 in T}.
[...] Read more.

A subgroup H of a topological abelian group X is said to be characterized by a sequence **v** = (v_{n}) of characters of *X* if *H* = {*x* ∈ *X* : v_{n}(*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.
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Locally Quasi-Convex Compatible Topologies on a Topological Group
**Abstract **

Received: 4 May 2015 / Revised: 28 September 2015 / Accepted: 8 October 2015 / Published: 13 October 2015

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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).
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Fixed Points of Local Actions of Lie Groups on Real and Complex 2-Manifolds
**Abstract **

Received: 30 April 2015 / Revised: 3 July 2015 / Accepted: 9 July 2015 / Published: 27 July 2015

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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.
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Pro-Lie Groups: A Survey with Open Problems
**Abstract **

Received: 17 June 2015 / Revised: 15 July 2015 / Accepted: 17 July 2015 / Published: 24 July 2015

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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.
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Lindelöf Σ-Spaces and R-Factorizable Paratopological Groups
**Abstract **

Received: 17 April 2015 / Revised: 25 June 2015 / Accepted: 30 June 2015 / Published: 10 July 2015

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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 cel_{w}
[...] 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 cel_{w}(*G*) ≤ w, and the Hewitt–Nachbin completion of *G* is again an R-factorizable paratopological group.
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On *T*-Characterized Subgroups of Compact Abelian Groups
**Abstract **

Received: 16 February 2015 / Revised: 11 June 2015 / Accepted: 16 June 2015 / Published: 19 June 2015

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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).
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Open and Dense Topological Transitivity of Extensions by Non-Compact Fiber of Hyperbolic Systems: A Review
**Abstract **

Received: 12 December 2014 / Revised: 9 January 2015 / Accepted: 26 January 2015 / Published: 4 February 2015

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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
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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.
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