What are relative open sets? - Mathematics Stack Exchange

Being open relative to X means that there is some open set O′ such that O=O′∩X, and equivalently for every x∈O there is some ε>0 such that Nε( ... MathematicsStackExchangeisaquestionandanswersiteforpeoplestudyingmathatanylevelandprofessionalsinrelatedfields.Itonlytakesaminutetosignup. Signuptojointhiscommunity Anybodycanaskaquestion Anybodycananswer Thebestanswersarevotedupandrisetothetop Home Public Questions Tags Users Unanswered Teams StackOverflowforTeams –Startcollaboratingandsharingorganizationalknowledge. CreateafreeTeam WhyTeams? Teams CreatefreeTeam Teams Q&Aforwork Connectandshareknowledgewithinasinglelocationthatisstructuredandeasytosearch. Learnmore Whatarerelativeopensets? AskQuestion Asked 9years,3monthsago Modified 3years,10monthsago Viewed 17ktimes 12 3 $\begingroup$ Icameacrossthefollowing: Definition15.Let$X$beasubsetof$\mathbb{R}$.Asubset$O\subsetX$issaidtobeopenin$X$(orrelativelyopenin$X$)ifforeach$x\inO$,thereexists$\epsilon=\epsilon(x)>0$suchthat$N_\epsilon(x)\capX\subsetO$. Whatis$\epsilon$and$N_\epsilon(x)$?Ormoregeneral,whatarerelativelyopensets? general-topologydefinition Share Cite Follow editedFeb6,2013at18:05 AsafKaragila♦ 370k4141goldbadges552552silverbadges948948bronzebadges askedFeb6,2013at17:18 onimonionimoni 5,64688goldbadges2626silverbadges4242bronzebadges $\endgroup$ 1 $\begingroup$ $N_{\epsilon}$isany(open)ballcontaining$x$ofradius$\epsilon$with$x$inthecenter. $\endgroup$ – user59671 Feb6,2013at17:25 Addacomment  |  2Answers 2 Sortedby: Resettodefault Highestscore(default) Datemodified(newestfirst) Datecreated(oldestfirst) 18 $\begingroup$ Forgetyourdefinitionabove.Thegeneralnotionis: Let$X$beatopologicalspace,$A\subsetX$anysubset.Aset$U_A$isrelativelyopenin$A$ifthereisanopenset$U$in$X$suchthat$U_A=U\capA$. Ithinkthatinyourdefinition$N_\epsilon(x)$ismeanttodenoteanopenneighborhoodofradius$\epsilon$of$x$,ie$(x-\epsilon,\x+\epsilon)$.Asyoucansee,thiswouldagreewiththedefinitionIgaveyouabove. Share Cite Follow answeredFeb6,2013at17:25 DanielRobert-NicoudDanielRobert-Nicoud 28.1k44goldbadges5959silverbadges123123bronzebadges $\endgroup$ 7 2 $\begingroup$ @Omar-Thedefinitionyougaveisspecifictometricspaces.Daniel'sdefinitionwillworkinanytopologicalspaceandhenceisofgreateruse. $\endgroup$ – ChrisLeary Feb6,2013at17:35 $\begingroup$ @DanielRobert-Nicoud-SupposeIwanttoprovesomething,e.g.continuity,suchthatIwouldneedtofindsomeopenset$U$.Woulditbesufficienttoshowthatthereexistsarelativelyopenset? $\endgroup$ – onimoni Feb6,2013at19:00 $\begingroup$ @omar:Whatwouldyouliketoproveexactly?GivemeamorespecificquestionandImight(should)beabletohelpyou. $\endgroup$ – DanielRobert-Nicoud Feb6,2013at19:07 $\begingroup$ @DanielRobert-Nicoud-InUrysohn'sMetrizationTheoremwehavesomespaceXandconstructafunction$F:X\rightarrowH$,where$H$isthehilbertcube.Theideaisthentoshowthat$F$isanembeddingbyproving$F$tobeone-to-one,continuousandopen.Whenprovingcontinuitylet$W$beopenin$X$,thentheyshowthat$F(W)$istheunionofrelativelyopensets.Sonormallywhenprovingcontinuityonewouldneedtofindanopen$F(W)$,butheretheysufficewiththeunionofrelativelyopensets. $\endgroup$ – onimoni Feb6,2013at20:18 $\begingroup$ @omar:Actuallytodowhatyousayistoprovethat$F$isanopenmap.Toproveitiscontinuousyoutakeanyrelativelyopensubset$U$ofitsimageandyouhavetoshowthatitspreimage$F^{-1}(U)$isopen. $\endgroup$ – DanielRobert-Nicoud Feb6,2013at20:28  |  Show2morecomments 7 $\begingroup$ Recallthatgenerally,$O$isopenifforevery$x\inO$thereexistssome$\varepsilon>0$suchthat$N_\varepsilon(x)\subseteqO$. Beingopenrelativeto$X$meansthatthereissomeopenset$O'$suchthat$O=O'\capX$,andequivalentlyforevery$x\inO$thereissome$\varepsilon>0$suchthat$N_\varepsilon(x)\capX\subseteqO$. Forexample$O=\{0\}$isnotopenin$\mathbbR$,butifweconsider$X=\{0\}$thenfor$\varepsilon=1$wehavethat$N_\varepsilon(0)\capX\subseteqO$,andthereforeitisopenrelativeto$X$. Share Cite Follow answeredFeb6,2013at17:41 AsafKaragila♦AsafKaragila 370k4141goldbadges552552silverbadges948948bronzebadges $\endgroup$ Addacomment  |  YourAnswer ThanksforcontributingananswertoMathematicsStackExchange!Pleasebesuretoanswerthequestion.Providedetailsandshareyourresearch!Butavoid…Askingforhelp,clarification,orrespondingtootheranswers.Makingstatementsbasedonopinion;backthemupwithreferencesorpersonalexperience.UseMathJaxtoformatequations.MathJaxreference.Tolearnmore,seeourtipsonwritinggreatanswers. Draftsaved Draftdiscarded Signuporlogin SignupusingGoogle SignupusingFacebook SignupusingEmailandPassword Submit Postasaguest Name Email Required,butnevershown PostYourAnswer Discard Byclicking“PostYourAnswer”,youagreetoourtermsofservice,privacypolicyandcookiepolicy Nottheansweryou'relookingfor?Browseotherquestionstaggedgeneral-topologydefinitionoraskyourownquestion. FeaturedonMeta AnnouncingthearrivalofValuedAssociate#1214:Dalmarus Improvementstositestatusandincidentcommunication Linked 1 $f$iscontinuousiff$f^{-1}(U)$isopenforallopen$U$. Related 8 WhatistheIdeaofaRelativeOpenSet? 1 OpensetsAreballs? 2 Open/ClosedSetsandrelatives 0 Provethatthesetsareopen 5 DefinitionofOpenset 1 proofverification,relativeopensets. 7 AreDisjointOpenSubsetsContainedinDisjointOpenSetsinaGeneralTopologicalSpace? 0 HowtoshowthatthefollowingdefinitionsofLebesguemeasurablesetsareequivalent. 0 Showingthatthefollowingareequivalentbyaproperroute. 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