{"title":"与拓扑空间相关的统计度量空间","authors":"B. Morrel, J. Nagata","doi":"10.1016/0016-660X(78)90026-0","DOIUrl":null,"url":null,"abstract":"<div><p>Our discussion answers the questions as to what topological spaces are statistically metrizable in the sense of Schweizer and Sklar [5] and whether this can be discerned by the <em>t</em>-norm on the space in the Menger triangle relation. Namely we prove (1) the class of topological Menger spaces coincides with that of semi-metrizable topological spaces, and (2) no condition weaker than 1=sup<sub><em>x</em><1</sub><em>T</em>(<em>x, x</em>) can guarantee that a Menger space satisfying the Menger triangle relation under <em>T</em> is topological.</p></div>","PeriodicalId":100574,"journal":{"name":"General Topology and its Applications","volume":"9 3","pages":"Pages 233-237"},"PeriodicalIF":0.0000,"publicationDate":"1978-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/0016-660X(78)90026-0","citationCount":"33","resultStr":"{\"title\":\"Statistical metric spaces as related to topological spaces\",\"authors\":\"B. Morrel, J. Nagata\",\"doi\":\"10.1016/0016-660X(78)90026-0\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Our discussion answers the questions as to what topological spaces are statistically metrizable in the sense of Schweizer and Sklar [5] and whether this can be discerned by the <em>t</em>-norm on the space in the Menger triangle relation. Namely we prove (1) the class of topological Menger spaces coincides with that of semi-metrizable topological spaces, and (2) no condition weaker than 1=sup<sub><em>x</em><1</sub><em>T</em>(<em>x, x</em>) can guarantee that a Menger space satisfying the Menger triangle relation under <em>T</em> is topological.</p></div>\",\"PeriodicalId\":100574,\"journal\":{\"name\":\"General Topology and its Applications\",\"volume\":\"9 3\",\"pages\":\"Pages 233-237\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1978-11-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/0016-660X(78)90026-0\",\"citationCount\":\"33\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"General Topology and its Applications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/0016660X78900260\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"General Topology and its Applications","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/0016660X78900260","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Statistical metric spaces as related to topological spaces
Our discussion answers the questions as to what topological spaces are statistically metrizable in the sense of Schweizer and Sklar [5] and whether this can be discerned by the t-norm on the space in the Menger triangle relation. Namely we prove (1) the class of topological Menger spaces coincides with that of semi-metrizable topological spaces, and (2) no condition weaker than 1=supx<1T(x, x) can guarantee that a Menger space satisfying the Menger triangle relation under T is topological.
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