Metrizability of Clifford topological semigroups
type: Item
instance of: scholarly article
title: Metrizability of Clifford topological semigroups
zbMATH Open document ID: 1250.22002
author: Taras Banakh
author: Oles Potiatynyk
author: O. Gutik
author: Oleksandr Ravs'kyj
published in: Semigroup Forum
publication date: 2012-08-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1105.2806
review text: By the Birkhoff-Kakutani theorem, a topological group \(G\) is metrizable if and only if it is first countable. In this paper, the authors prove that a countably compact Clifford topological semigroup \(S\) is metrizable if and only if the set \(E\) of idempotents of \(S\) is a metrizable \(G_\delta\)-set in \(S\). The proof is divided into a series of the following claims: \textbf{Claim 1:} The space \(S\) is first countable at each point \(e\in E\). \textbf{Claim 2:} The inversion \(x\mapsto x^{-1}:S\rightarrow S\) is continuous at each \(e\in E\). \textbf{Claim 3:} For every idempotent \(e\in E\) the maximal subgroup \(H_e=\{x\in S:xx^{-1}=e\}\) is a metrizable topological group. \textbf{Claim 4:} The semigroup \(S\) is topologically periodic, i.e., if for each element \(x\in S\) and each neighborhood \(O_x\subseteq S\) of \(x\) there is an integer \(n\geq 2\) with \(x^n\in O_x\). \textbf{Claim 5:} \(S\) is a topological Clifford semigroup.
Mathematics Subject Classification ID: 22A15
zbMATH DE Number: 6066640
zbMATH Keywords: Clifford semigroup
zbMATH Keywords: metrizable topological semigroup
zbMATH Keywords: topologically periodic semigroup
zbMATH Keywords: countably compact space
reviewed by: M. Akbari Tootkaboni
MaRDI profile type: Publication
OpenAlex ID: W2089878315
arXiv ID: 1105.2806
cites work: On cardinal invariants and metrizability of topological inverse Clifford semigroups
cites work: On the continuity of inversion in countably compact inverse topological semigroups
cites work: Sequentially compact Hausdorff cancellative semigroup is a topological group
cites work: The continuity of the inversion and the structure of maximal subgroups in countably compact topological semigroups
cites work: Q5507672
cites work: Products of normal spaces with metric spaces
DOI: 10.1007/S00233-011-9341-7
description: scientific article; zbMATH DE number 6066640