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12 pages, 332 KiB

Open AccessArticle

Factoring Continuous Characters Defined on Subgroups of Products of Topological Groups

by Mikhail G. Tkachenko

Axioms 2021, 10(3), 167; https://doi.org/10.3390/axioms10030167 - 28 Jul 2021

Viewed by 1868

This study is on the factorization properties of continuous homomorphisms defined on subgroups (or submonoids) of products of (para)topological groups (or monoids). A typical result is the following one: Let

D = \prod_{i \in I} D_{i}

be a product of paratopological [...] Read more.

D = \prod_{i \in I} D_{i}

be a product of paratopological groups, S be a dense subgroup of D, and

χ

a continuous character of S. Then one can find a finite set

E \subset I

and continuous characters

χ_{i}

D_{i}

, for

i \in E

, such that

χ = (\prod_{i \in E} χ_{i} \circ p_{i}) S

, where

p_{i} : D \to D_{i}

is the projection. Full article

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

33 pages, 433 KiB

Open AccessArticle

Categorically Closed Topological Groups

by Taras Banakh

Axioms 2017, 6(3), 23; https://doi.org/10.3390/axioms6030023 - 30 Jul 2017

Cited by 12 | Viewed by 5235

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 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

(This article belongs to the Collection Topological Groups)

14 pages, 235 KiB

Open AccessArticle

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 3 | Viewed by 4703

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 cel_w [...] Read more.

(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)

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