{"title":"一类非微不足道的非 D 空间简单示例","authors":"Yu-Lin Chou","doi":"10.1515/gmj-2024-2033","DOIUrl":null,"url":null,"abstract":"Given any regular <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>T</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2033_eq_0029.png\"/> <jats:tex-math>{T_{0}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (equivalently, regular <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>T</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2033_eq_0030.png\"/> <jats:tex-math>{T_{1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>) space <jats:italic>X</jats:italic>, the question of whether <jats:italic>X</jats:italic> being Lindelöf implies <jats:italic>X</jats:italic> being a <jats:italic>D</jats:italic>-space is an active open problem. This article gives a class of handy examples of a second countable collectionwise normal collectionwise Hausdorff <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>T</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2033_eq_0029.png\"/> <jats:tex-math>{T_{0}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> space of uncountable cardinal, with at most countably many singletons being not closed, that is not a <jats:italic>D</jats:italic>-space. Also given is a class of handy examples of a second countable hyperconnected <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>T</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2033_eq_0029.png\"/> <jats:tex-math>{T_{0}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> space of uncountable cardinal, with at most countably many singletons being not closed, that is not a <jats:italic>D</jats:italic>-space.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A class of nontrivial simple examples of a non-D-space\",\"authors\":\"Yu-Lin Chou\",\"doi\":\"10.1515/gmj-2024-2033\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Given any regular <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>T</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2024-2033_eq_0029.png\\\"/> <jats:tex-math>{T_{0}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (equivalently, regular <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>T</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2024-2033_eq_0030.png\\\"/> <jats:tex-math>{T_{1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>) space <jats:italic>X</jats:italic>, the question of whether <jats:italic>X</jats:italic> being Lindelöf implies <jats:italic>X</jats:italic> being a <jats:italic>D</jats:italic>-space is an active open problem. This article gives a class of handy examples of a second countable collectionwise normal collectionwise Hausdorff <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>T</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2024-2033_eq_0029.png\\\"/> <jats:tex-math>{T_{0}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> space of uncountable cardinal, with at most countably many singletons being not closed, that is not a <jats:italic>D</jats:italic>-space. Also given is a class of handy examples of a second countable hyperconnected <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>T</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2024-2033_eq_0029.png\\\"/> <jats:tex-math>{T_{0}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> space of uncountable cardinal, with at most countably many singletons being not closed, that is not a <jats:italic>D</jats:italic>-space.\",\"PeriodicalId\":55101,\"journal\":{\"name\":\"Georgian Mathematical Journal\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-06-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Georgian Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/gmj-2024-2033\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Georgian Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/gmj-2024-2033","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
给定任何正则 T 0 {T_{0}} （等价于正则 T 1 {T_{1}} ）空间 X (等价地，正则 T 1 {T_{1}} ）空间 X，X 是林德洛夫是否意味着 X 是 D 空间是一个活跃的开放问题。本文给出了一类非 D 空间的第二可数集合正则集合 Hausdorff T 0 {T_{0}}空间的方便例子，该空间具有最多可数的单子不封闭。此外，还给出了一类非 D 空间的第二可数超连接 T 0 {T_{0}} 空间的方便示例，该空间具有最多可数个不封闭的单子。
A class of nontrivial simple examples of a non-D-space
Given any regular T0{T_{0}} (equivalently, regular T1{T_{1}}) space X, the question of whether X being Lindelöf implies X being a D-space is an active open problem. This article gives a class of handy examples of a second countable collectionwise normal collectionwise Hausdorff T0{T_{0}} space of uncountable cardinal, with at most countably many singletons being not closed, that is not a D-space. Also given is a class of handy examples of a second countable hyperconnected T0{T_{0}} space of uncountable cardinal, with at most countably many singletons being not closed, that is not a D-space.
期刊介绍：
The Georgian Mathematical Journal was founded by the Georgian National Academy of Sciences and A. Razmadze Mathematical Institute, and is jointly produced with De Gruyter. The concern of this international journal is the publication of research articles of best scientific standard in pure and applied mathematics. Special emphasis is put on the presentation of results obtained by Georgian mathematicians.
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