{"title":"On some kinds of ω-balancedness and (*) properties in certain semitopological groups","authors":"Liang-Xue Peng","doi":"10.1016/j.topol.2024.109001","DOIUrl":null,"url":null,"abstract":"<div><p>In this article, we discuss some relationships of <em>ω</em>-balancedness and <span><math><mo>(</mo><mo>⁎</mo><mo>)</mo></math></span> properties which were introduced for giving characterizations of subgroups of topological products of certain para(semi)topological groups. We mainly get the following results.</p><p>If <em>G</em> is a regular <em>ω</em>-balanced locally <em>ω</em>-good semitopological group with a <em>q</em>-point, then <span><math><mi>I</mi><mi>r</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span> if and only if <span><math><mi>S</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span>. If <em>G</em> is a regular strongly paracompact semitopological group with a <em>q</em>-point and <span><math><mi>S</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span>, then <em>G</em> is completely <em>ω</em>-balanced if and only if <em>G</em> has property <span><math><mo>(</mo><msup><mrow></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></math></span>. If <em>G</em> is a regular paracompact <em>ω</em>-balanced locally good semitopological group with a <em>q</em>-point and <span><math><mi>S</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span>, then <em>G</em> has property <span><math><mo>(</mo><mi>w</mi><mo>⁎</mo><mo>)</mo></math></span> if and only if <em>G</em> has property (**). If <em>G</em> is a regular metacompact semitopological group with a <em>q</em>-point and <span><math><mi>S</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span>, then <em>G</em> is <em>MM</em>-<em>ω</em>-balanced if and only if <em>G</em> is <em>M</em>-<em>ω</em>-balanced.</p><p>We show that a semitopological group <em>G</em> admits a homeomorphic embedding as a subgroup of a product of metrizable semitopological groups if and only if <em>G</em> is topologically isomorphic to a subgroup of a product of semitopological groups which are first-countable paracompact regular <em>σ</em>-spaces and is topologically isomorphic to a subgroup of a product of Moore semitopological groups.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-06-18","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/S016686412400186X","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 this article, we discuss some relationships of ω-balancedness and properties which were introduced for giving characterizations of subgroups of topological products of certain para(semi)topological groups. We mainly get the following results.
If G is a regular ω-balanced locally ω-good semitopological group with a q-point, then if and only if . If G is a regular strongly paracompact semitopological group with a q-point and , then G is completely ω-balanced if and only if G has property . If G is a regular paracompact ω-balanced locally good semitopological group with a q-point and , then G has property if and only if G has property (**). If G is a regular metacompact semitopological group with a q-point and , then G is MM-ω-balanced if and only if G is M-ω-balanced.
We show that a semitopological group G admits a homeomorphic embedding as a subgroup of a product of metrizable semitopological groups if and only if G is topologically isomorphic to a subgroup of a product of semitopological groups which are first-countable paracompact regular σ-spaces and is topologically isomorphic to a subgroup of a product of Moore semitopological groups.
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.