{"title":"属于D空间的Box和nabla产品","authors":"H.A. Barriga-Acosta , P.M. Gartside","doi":"10.1016/j.indag.2023.04.002","DOIUrl":null,"url":null,"abstract":"<div><p>A space <span><math><mi>X</mi></math></span> is <span><math><mi>D</mi></math></span> if for every assignment, <span><math><mi>U</mi></math></span><span>, of an open neighborhood to each point </span><span><math><mi>x</mi></math></span> in <span><math><mi>X</mi></math></span> there is a closed discrete <span><math><mi>D</mi></math></span> such that <span><math><mrow><mo>⋃</mo><mrow><mo>{</mo><mi>U</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>:</mo><mi>x</mi><mo>∈</mo><mi>D</mi><mo>}</mo></mrow><mo>=</mo><mi>X</mi></mrow></math></span>. The box product, <span><math><mrow><mo>□</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span>, is <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span> with topology generated by all <span><math><mrow><msub><mrow><mo>∏</mo></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mi>U</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>, where every <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is open. The nabla product, <span><math><mrow><mo>∇</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span>, is obtained from <span><math><mrow><mo>□</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span> by quotienting out mod-finite. The weight of <span><math><mi>X</mi></math></span>, <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span>, is the minimal size of a base, while <span><math><mrow><mi>d</mi><mo>=</mo><mo>cof</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span>.</p><p>It is shown that there are specific compact spaces <span><math><mi>X</mi></math></span> such that <span><math><mrow><mo>□</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span> and <span><math><mrow><mo>∇</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span> are not <span><math><mi>D</mi></math></span>, but in general:</p><p>(1) <span><math><mrow><mo>□</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span> and <span><math><mrow><mo>∇</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span> are hereditarily <span><math><mi>D</mi></math></span> if <span><math><mi>X</mi></math></span> is scattered and either hereditarily paracompact or of finite scattered height, or if <span><math><mi>X</mi></math></span> is metrizable (and <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>≤</mo><mi>d</mi></mrow></math></span> for <span><math><mrow><mo>□</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span>);</p><p>(2) <span><math><mrow><mo>∇</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span> is hereditarily <span><math><mi>D</mi></math></span> if <span><math><mi>X</mi></math></span> is first countable and <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>≤</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>, or <em>consistently</em> if <span><math><mi>X</mi></math></span> is first countable and <span><math><mrow><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><mi>c</mi></mrow></math></span>, or <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>≤</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>; and</p><p>(3) <span><math><mrow><mo>□</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span> is <span><math><mi>D</mi></math></span> <em>consistently</em> if <span><math><mi>X</mi></math></span> is compact and either first countable or <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>≤</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"34 6","pages":"Pages 1237-1253"},"PeriodicalIF":0.5000,"publicationDate":"2023-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Box and nabla products that are D-spaces\",\"authors\":\"H.A. Barriga-Acosta , P.M. Gartside\",\"doi\":\"10.1016/j.indag.2023.04.002\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A space <span><math><mi>X</mi></math></span> is <span><math><mi>D</mi></math></span> if for every assignment, <span><math><mi>U</mi></math></span><span>, of an open neighborhood to each point </span><span><math><mi>x</mi></math></span> in <span><math><mi>X</mi></math></span> there is a closed discrete <span><math><mi>D</mi></math></span> such that <span><math><mrow><mo>⋃</mo><mrow><mo>{</mo><mi>U</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>:</mo><mi>x</mi><mo>∈</mo><mi>D</mi><mo>}</mo></mrow><mo>=</mo><mi>X</mi></mrow></math></span>. The box product, <span><math><mrow><mo>□</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span>, is <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span> with topology generated by all <span><math><mrow><msub><mrow><mo>∏</mo></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mi>U</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>, where every <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is open. The nabla product, <span><math><mrow><mo>∇</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span>, is obtained from <span><math><mrow><mo>□</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span> by quotienting out mod-finite. The weight of <span><math><mi>X</mi></math></span>, <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span>, is the minimal size of a base, while <span><math><mrow><mi>d</mi><mo>=</mo><mo>cof</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span>.</p><p>It is shown that there are specific compact spaces <span><math><mi>X</mi></math></span> such that <span><math><mrow><mo>□</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span> and <span><math><mrow><mo>∇</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span> are not <span><math><mi>D</mi></math></span>, but in general:</p><p>(1) <span><math><mrow><mo>□</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span> and <span><math><mrow><mo>∇</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span> are hereditarily <span><math><mi>D</mi></math></span> if <span><math><mi>X</mi></math></span> is scattered and either hereditarily paracompact or of finite scattered height, or if <span><math><mi>X</mi></math></span> is metrizable (and <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>≤</mo><mi>d</mi></mrow></math></span> for <span><math><mrow><mo>□</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span>);</p><p>(2) <span><math><mrow><mo>∇</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span> is hereditarily <span><math><mi>D</mi></math></span> if <span><math><mi>X</mi></math></span> is first countable and <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>≤</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>, or <em>consistently</em> if <span><math><mi>X</mi></math></span> is first countable and <span><math><mrow><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><mi>c</mi></mrow></math></span>, or <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>≤</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>; and</p><p>(3) <span><math><mrow><mo>□</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></math></span> is <span><math><mi>D</mi></math></span> <em>consistently</em> if <span><math><mi>X</mi></math></span> is compact and either first countable or <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>≤</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>.</p></div>\",\"PeriodicalId\":56126,\"journal\":{\"name\":\"Indagationes Mathematicae-New Series\",\"volume\":\"34 6\",\"pages\":\"Pages 1237-1253\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2023-05-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Indagationes Mathematicae-New Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0019357723000381\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Indagationes Mathematicae-New Series","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0019357723000381","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
A space is if for every assignment, , of an open neighborhood to each point in there is a closed discrete such that . The box product, , is with topology generated by all , where every is open. The nabla product, , is obtained from by quotienting out mod-finite. The weight of , , is the minimal size of a base, while .
It is shown that there are specific compact spaces such that and are not , but in general:
(1) and are hereditarily if is scattered and either hereditarily paracompact or of finite scattered height, or if is metrizable (and for );
(2) is hereditarily if is first countable and , or consistently if is first countable and , or ; and
(3) is consistently if is compact and either first countable or .
期刊介绍:
Indagationes Mathematicae is a peer-reviewed international journal for the Mathematical Sciences of the Royal Dutch Mathematical Society. The journal aims at the publication of original mathematical research papers of high quality and of interest to a large segment of the mathematics community. The journal also welcomes the submission of review papers of high quality.