{"title":"Products of two sober dcpo's need not be sober","authors":"Hualin Miao, Xiaoyong Xi, Xiaodong Jia, Qingguo Li, Dongsheng Zhao","doi":"arxiv-2408.08587","DOIUrl":null,"url":null,"abstract":"We constructed two dcpo's whose Scott spaces are sober, but the Scott space\nof their order product is not sober. This answers an open problem on the\nsobriety of Scott spaces. Meantime, we show that if $M$ and $N$ are special\ntype of sober complete lattices, then the Scott space of their order product\n$M\\times N$ is sober.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"19 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-16","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-2408.08587","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We constructed two dcpo's whose Scott spaces are sober, but the Scott space
of their order product is not sober. This answers an open problem on the
sobriety of Scott spaces. Meantime, we show that if $M$ and $N$ are special
type of sober complete lattices, then the Scott space of their order product
$M\times N$ is sober.
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