{"title":"A completeness property of some function spaces","authors":"Surjit Singh Khurana","doi":"10.1016/0016-660X(78)90027-2","DOIUrl":null,"url":null,"abstract":"<div><p>It is proved that if <em>X</em> is a sequentially compact Hausdorff space, <em>E</em> a Hausdorff complete uniform space, <em>C</em>(<em>X, E</em>) the space of all <em>E</em>-valued continuous functions on <em>X</em> with <span><math><mtext>C</mtext></math></span> uniformity, <span><math><mtext>C</mtext></math></span> being the class of all compact subsets of <em>X</em>, and <em>H</em> a closed subset of <em>C</em>(<em>X, E</em>) containing a countable union of precompact subsets of <em>C</em>(<em>X, E</em>) as a dense subset, then <em>H</em> is complete.</p></div>","PeriodicalId":100574,"journal":{"name":"General Topology and its Applications","volume":"9 3","pages":"Pages 239-241"},"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)90027-2","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"General Topology and its Applications","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/0016660X78900272","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
It is proved that if X is a sequentially compact Hausdorff space, E a Hausdorff complete uniform space, C(X, E) the space of all E-valued continuous functions on X with uniformity, being the class of all compact subsets of X, and H a closed subset of C(X, E) containing a countable union of precompact subsets of C(X, E) as a dense subset, then H is complete.
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