Relatively-open (-closed) set - Encyclopedia of Mathematics

A set M in a topological space is relatively open (relatively closed) with respect to E if and only if M∩E is open (respectively, closed) in E ...   Login www.springer.com TheEuropeanMathematicalSociety Navigation Mainpage PagesA-Z StatProbCollection Recentchanges Currentevents Randompage Help Projecttalk Requestaccount Tools Whatlinkshere Relatedchanges Specialpages Printableversion Permanentlink Pageinformation Namespaces Page Discussion Variants Views View Viewsource History Actions Relatively-open(-closed)set FromEncyclopediaofMathematics Jumpto:navigation, search setopen(closed)relative(orwithrespectto)toacertainset$E$inatopologicalspace$X$" Aset$M$in$X$suchthat $$ M=E\setminus\overline{(E\setminusM)}\,\\(\,M=E\cap\barM\,) $$ (thebardenotestheoperationofclosure,cf.Closureofaset).Foracertainsettobeopen(closed)relativeto$E$,itisnecessaryandsufficientthatitistheintersectionof$E$andacertainopen(closed)set. Comments Aset$M$inatopologicalspaceisrelativelyopen(relativelyclosed)withrespectto$E$ifandonlyif$M\capE$isopen(respectively,closed)in$E$fortherelativetopologyon$E$. References [a1]P.S.[P.S.Aleksandrov]Alexandroff,H.Hopf,"Topologie",Chelsea,reprint(1972)pp.33ff,44ff [a2]C.Kuratowski,"Introductiontosettheoryandtopology",Pergamon(1961)pp.128ff(TranslatedfromFrench) HowtoCiteThisEntry:Relatively-open(-closed)set.EncyclopediaofMathematics.URL:http://encyclopediaofmath.org/index.php?title=Relatively-open_(-closed)_set&oldid=34421ThisarticlewasadaptedfromanoriginalarticlebyM.I.Voitsekhovskii(originator),whichappearedinEncyclopediaofMathematics-ISBN1402006098.Seeoriginalarticle Retrievedfrom"https://encyclopediaofmath.org/index.php?title=Relatively-open_(-closed)_set&oldid=34421" Categories:TeXdoneGeneraltopology Thispagewaslasteditedon9November2014,at13:57. Privacypolicy AboutEncyclopediaofMathematics Disclaimers Copyrights Impressum-Legal ManageCookies