{"title":"On MM-ω-balancedness and FR(Fm)-factorizable semi(para)topological groups","authors":"Liang-Xue Peng, Yu-Ming Deng","doi":"10.1016/j.topol.2024.109183","DOIUrl":null,"url":null,"abstract":"<div><div>In the second part of this article, we introduce a notion which is called <em>MM</em>-<em>ω</em>-balancedness in the class of semitopological groups. We show that if <em>G</em> is a semitopological (paratopological) group, then <em>G</em> is topologically isomorphic to a subgroup of the product of a family of metacompact Moore semitopological (paratopological) groups if and only if <em>G</em> is regular <em>MM</em>-<em>ω</em>-balanced and <span><math><mi>I</mi><mi>r</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span>. If <em>G</em> is a <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> <em>bM</em>-<em>ω</em>-balanced semitopological group and <span><math><mi>f</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>H</mi></math></span> is an open continuous homomorphism of <em>G</em> onto a first-countable semitopological group <em>H</em> such that <span><math><mi>ker</mi><mo></mo><mo>(</mo><mi>f</mi><mo>)</mo></math></span> is a countably compact subgroup of <em>G</em>, then <em>H</em> is a metacompact developable space.</div><div>In the third part of this article, we introduce notions of <span><math><mi>F</mi><mi>R</mi></math></span>-factorizability and <span><math><mi>F</mi><mi>m</mi></math></span>-factorizability. We give some equivalent conditions that a semitopological (paratopological) group is <span><math><mi>F</mi><mi>R</mi></math></span>-factorizable or <span><math><mi>F</mi><mi>m</mi></math></span>-factorizable. If <em>G</em> is a Tychonoff <span><math><mi>F</mi><mi>R</mi></math></span> (<span><math><mi>F</mi><mi>m</mi></math></span>)-factorizable semitopological group and <span><math><mi>f</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>H</mi></math></span> is a continuous open homomorphism of <em>G</em> onto a semitopological group <em>H</em>, then <em>H</em> is <span><math><mi>F</mi><mi>R</mi></math></span> (<span><math><mi>F</mi><mi>m</mi></math></span>)-factorizable. If <em>G</em> is a <span><math><mi>F</mi><mi>R</mi></math></span> (<span><math><mi>F</mi><mi>m</mi></math></span>)-factorizable paratopological group and <span><math><mi>f</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>H</mi></math></span> is a continuous <em>d</em>-open homomorphism of <em>G</em> onto a paratopological group <em>H</em>, then <em>H</em> is <span><math><mi>F</mi><mi>R</mi></math></span> (<span><math><mi>F</mi><mi>m</mi></math></span>)-factorizable.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"360 ","pages":"Article 109183"},"PeriodicalIF":0.6000,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Topology and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166864124003687","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In the second part of this article, we introduce a notion which is called MM-ω-balancedness in the class of semitopological groups. We show that if G is a semitopological (paratopological) group, then G is topologically isomorphic to a subgroup of the product of a family of metacompact Moore semitopological (paratopological) groups if and only if G is regular MM-ω-balanced and . If G is a bM-ω-balanced semitopological group and is an open continuous homomorphism of G onto a first-countable semitopological group H such that is a countably compact subgroup of G, then H is a metacompact developable space.
In the third part of this article, we introduce notions of -factorizability and -factorizability. We give some equivalent conditions that a semitopological (paratopological) group is -factorizable or -factorizable. If G is a Tychonoff ()-factorizable semitopological group and is a continuous open homomorphism of G onto a semitopological group H, then H is ()-factorizable. If G is a ()-factorizable paratopological group and is a continuous d-open homomorphism of G onto a paratopological group H, then H is ()-factorizable.
期刊介绍:
Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology.
At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.