关于ZF$\mathsf｛ZF｝中的Hausdorff算子$

Pub Date : 2023-07-24 DOI:10.1002/malq.202300004

Kyriakos Keremedis, Eleftherios Tachtsis

{"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":"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 F—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 U and V are disjoint open neighborhoods of x and y, 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>):\n ","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"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\":\"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 F—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 U and V are disjoint open neighborhoods of x and y, 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>):\\n \",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-07-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/malq.202300004\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/malq.202300004","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

摘要

如果存在称为豪斯多夫算子的函数F，则豪斯多夫空间（X，T）$（X，\mathcal｛T｝）$被有效地称为豪斯道夫，使得对于每个X，y∈X$X，y\in X$中X≠y$X\ne y$，F（x，y）=（U，V）$F（x）=（U，V）$，其中U和V分别是x和y的不相交的开邻域。在其他结果中，我们在ZF$\mathsf{ZF}$中，即在没有选择公理的Zermelo–Fraenkel集合论（AC$\mathsf{AC}$）中建立了以下结果：

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文

On Hausdorff operators in ZF $\mathsf {ZF}$

A Hausdorff space $(X, T)$ is called effectively Hausdorff if there exists a function F—called a Hausdorff operator—such that, for every $x, y \in X$ with $x \neq y$ , $F (x, y) = (U, V)$ , where U and V are disjoint open neighborhoods of x and y, respectively. Among other results, we establish the following in $ZF$ , i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice ( $AC$ ):

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助