**Abstract**

# Topical Collection "Topological Groups"

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

## Editor

2. School of Engineering and Mathematical Sciences, La Trobe University, Bundoora, Victoria 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; 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

Special Issues and Collections in MDPI journals

## 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*

**Manuscript Submission Information**

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the collection website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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 1400 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.

## 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

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**Abstract**

*p*-groups (

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*p*-groups (

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*Structure Theorem for Protori*is derived for the category of finite-dimensional

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*Structure Theorem for Protori*is derived for the category of finite-dimensional

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*G*in the category of

*k*-groups. All totally disconnected locally compact groups are

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*K*[...] Read more.

*K*; and

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*A*, a topological group

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*c*

_{0}

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*S*and

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*G*and demonstrate some applications using Topological Groups, Model Theory, Geometric Group Theory, and Topological Dynamics. [...] Read more.

*G*and demonstrate some applications using Topological Groups, Model Theory, Geometric Group Theory, and Topological Dynamics. Full article

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**Abstract**

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*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

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^{|K|}-many proper dense pseudocompact subgroups. (B) (2003) Every non-metrizable compact abelian group K admits 2

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^{|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. Full article

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**v**= (v

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*X*: v

_{n}(

*x*) → 0 in T}. [...] Read more.

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_{n}) of characters of

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*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. Full article

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*G*) that are compatible with (τ) (

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*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

**Abstract**

**Abstract**

*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,

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*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,

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*T*-Characterized Subgroups of Compact Abelian Groups Cited by 4

**Abstract**

*Topol. Appl.*

**2013**,

*160*, 2427–2442). Full article

**Abstract**