{"title":"关于对角线度和星形网络","authors":"Nathan Carlson","doi":"arxiv-2407.13508","DOIUrl":null,"url":null,"abstract":"Given an open cover $\\mathcal{U}$ of a topological space $X$, we introduce\nthe notion of a star network for $\\mathcal{U}$. The associated cardinal\nfunction $sn(X)$, where $e(X)\\leq sn(X)\\leq L(X)$, is used to establish new\ncardinal inequalities involving diagonal degrees. We show $|X|\\leq\nsn(X)^{\\Delta(X)}$ for a $T_1$ space $X$, giving a partial answer to a\nlong-standing question of Angelo Bella. Many further results are given using\nvariations of $sn(X)$. One result has as corollaries Buzyakova's theorem that a\nccc space with a regular $G_\\delta$-diagonal has cardinality at most\n$\\mathfrak{c}$, as well as three results of Gotchev. Further results lead to\nlogical improvements of theorems of Basile, Bella, and Ridderbos, a partial\nsolution to a question of the same authors, and a theorem of Gotchev,\nTkachenko, and Tkachuk. Finally, we define the Urysohn extent $Ue(X)$ with the\nproperty $Ue(X)\\leq\\min\\{aL(X),e(X)\\}$ and use the Erd\\H{o}s-Rado theorem to\nshow that $|X|\\leq 2^{Ue(X)\\overline{\\Delta}(X)}$ for any Urysohn space $X$.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"16 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On diagonal degrees and star networks\",\"authors\":\"Nathan Carlson\",\"doi\":\"arxiv-2407.13508\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Given an open cover $\\\\mathcal{U}$ of a topological space $X$, we introduce\\nthe notion of a star network for $\\\\mathcal{U}$. The associated cardinal\\nfunction $sn(X)$, where $e(X)\\\\leq sn(X)\\\\leq L(X)$, is used to establish new\\ncardinal inequalities involving diagonal degrees. We show $|X|\\\\leq\\nsn(X)^{\\\\Delta(X)}$ for a $T_1$ space $X$, giving a partial answer to a\\nlong-standing question of Angelo Bella. Many further results are given using\\nvariations of $sn(X)$. One result has as corollaries Buzyakova's theorem that a\\nccc space with a regular $G_\\\\delta$-diagonal has cardinality at most\\n$\\\\mathfrak{c}$, as well as three results of Gotchev. Further results lead to\\nlogical improvements of theorems of Basile, Bella, and Ridderbos, a partial\\nsolution to a question of the same authors, and a theorem of Gotchev,\\nTkachenko, and Tkachuk. Finally, we define the Urysohn extent $Ue(X)$ with the\\nproperty $Ue(X)\\\\leq\\\\min\\\\{aL(X),e(X)\\\\}$ and use the Erd\\\\H{o}s-Rado theorem to\\nshow that $|X|\\\\leq 2^{Ue(X)\\\\overline{\\\\Delta}(X)}$ for any Urysohn space $X$.\",\"PeriodicalId\":501314,\"journal\":{\"name\":\"arXiv - MATH - General Topology\",\"volume\":\"16 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-07-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - General Topology\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2407.13508\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - General Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.13508","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Given an open cover $\mathcal{U}$ of a topological space $X$, we introduce
the notion of a star network for $\mathcal{U}$. The associated cardinal
function $sn(X)$, where $e(X)\leq sn(X)\leq L(X)$, is used to establish new
cardinal inequalities involving diagonal degrees. We show $|X|\leq
sn(X)^{\Delta(X)}$ for a $T_1$ space $X$, giving a partial answer to a
long-standing question of Angelo Bella. Many further results are given using
variations of $sn(X)$. One result has as corollaries Buzyakova's theorem that a
ccc space with a regular $G_\delta$-diagonal has cardinality at most
$\mathfrak{c}$, as well as three results of Gotchev. Further results lead to
logical improvements of theorems of Basile, Bella, and Ridderbos, a partial
solution to a question of the same authors, and a theorem of Gotchev,
Tkachenko, and Tkachuk. Finally, we define the Urysohn extent $Ue(X)$ with the
property $Ue(X)\leq\min\{aL(X),e(X)\}$ and use the Erd\H{o}s-Rado theorem to
show that $|X|\leq 2^{Ue(X)\overline{\Delta}(X)}$ for any Urysohn space $X$.