{"title":"关于ZF$\\mathsf｛ZF｝中的Hausdorff算子$","authors":"Kyriakos Keremedis, Eleftherios Tachtsis","doi":"10.1002/malq.202300004","DOIUrl":null,"url":null,"abstract":"<p>A Hausdorff space <math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>X</mi>\n <mo>,</mo>\n <mi>T</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$(X,\\mathcal {T})$</annotation>\n </semantics></math> is called effectively Hausdorff if there exists a function <i>F</i>—called a Hausdorff operator—such that, for every <math>\n <semantics>\n <mrow>\n <mi>x</mi>\n <mo>,</mo>\n <mi>y</mi>\n <mo>∈</mo>\n <mi>X</mi>\n </mrow>\n <annotation>$x,y\\in X$</annotation>\n </semantics></math> with <math>\n <semantics>\n <mrow>\n <mi>x</mi>\n <mo>≠</mo>\n <mi>y</mi>\n </mrow>\n <annotation>$x\\ne y$</annotation>\n </semantics></math>, <math>\n <semantics>\n <mrow>\n <mi>F</mi>\n <mo>(</mo>\n <mi>x</mi>\n <mo>,</mo>\n <mi>y</mi>\n <mo>)</mo>\n <mo>=</mo>\n <mo>(</mo>\n <mi>U</mi>\n <mo>,</mo>\n <mi>V</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$F(x,y)=(U,V)$</annotation>\n </semantics></math>, where <i>U</i> and <i>V</i> are disjoint open neighborhoods of <i>x</i> and <i>y</i>, respectively. Among other results, we establish the following in <math>\n <semantics>\n <mi>ZF</mi>\n <annotation>$\\mathsf {ZF}$</annotation>\n </semantics></math>, i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice (<math>\n <semantics>\n <mi>AC</mi>\n <annotation>$\\mathsf {AC}$</annotation>\n </semantics></math>):</p><p>\n </p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 3","pages":"347-369"},"PeriodicalIF":0.4000,"publicationDate":"2023-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Hausdorff operators in \\n \\n ZF\\n $\\\\mathsf {ZF}$\",\"authors\":\"Kyriakos Keremedis, Eleftherios Tachtsis\",\"doi\":\"10.1002/malq.202300004\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A Hausdorff space <math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mi>X</mi>\\n <mo>,</mo>\\n <mi>T</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(X,\\\\mathcal {T})$</annotation>\\n </semantics></math> is called effectively Hausdorff if there exists a function <i>F</i>—called a Hausdorff operator—such that, for every <math>\\n <semantics>\\n <mrow>\\n <mi>x</mi>\\n <mo>,</mo>\\n <mi>y</mi>\\n <mo>∈</mo>\\n <mi>X</mi>\\n </mrow>\\n <annotation>$x,y\\\\in X$</annotation>\\n </semantics></math> with <math>\\n <semantics>\\n <mrow>\\n <mi>x</mi>\\n <mo>≠</mo>\\n <mi>y</mi>\\n </mrow>\\n <annotation>$x\\\\ne y$</annotation>\\n </semantics></math>, <math>\\n <semantics>\\n <mrow>\\n <mi>F</mi>\\n <mo>(</mo>\\n <mi>x</mi>\\n <mo>,</mo>\\n <mi>y</mi>\\n <mo>)</mo>\\n <mo>=</mo>\\n <mo>(</mo>\\n <mi>U</mi>\\n <mo>,</mo>\\n <mi>V</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$F(x,y)=(U,V)$</annotation>\\n </semantics></math>, where <i>U</i> and <i>V</i> are disjoint open neighborhoods of <i>x</i> and <i>y</i>, respectively. Among other results, we establish the following in <math>\\n <semantics>\\n <mi>ZF</mi>\\n <annotation>$\\\\mathsf {ZF}$</annotation>\\n </semantics></math>, i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice (<math>\\n <semantics>\\n <mi>AC</mi>\\n <annotation>$\\\\mathsf {AC}$</annotation>\\n </semantics></math>):</p><p>\\n </p>\",\"PeriodicalId\":49864,\"journal\":{\"name\":\"Mathematical Logic Quarterly\",\"volume\":\"69 3\",\"pages\":\"347-369\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2023-07-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Logic Quarterly\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/malq.202300004\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"LOGIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Logic Quarterly","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/malq.202300004","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"LOGIC","Score":null,"Total":0}
A Hausdorff space is called effectively Hausdorff if there exists a function F—called a Hausdorff operator—such that, for every with , , where U and V are disjoint open neighborhoods of x and y, respectively. Among other results, we establish the following in , i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice ():
期刊介绍：
Mathematical Logic Quarterly publishes original contributions on mathematical logic and foundations of mathematics and related areas, such as general logic, model theory, recursion theory, set theory, proof theory and constructive mathematics, algebraic logic, nonstandard models, and logical aspects of theoretical computer science.
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