[Real Analysis] Definition of relatively open set : r/learnmath
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[Real Analysis] Definition of relatively open set · Each set in the subspace topology is actually a subset of S · The empty set and S are both ... Foundtheinternet!2[RealAnalysis]Definitionofrelativelyopenset Close2Postedbyu/[deleted]4yearsago[RealAnalysis]Definitionofrelativelyopenset Hi!I'mhavingtroubleunderstandingthefollowingdefinition:AsetVCSCRnisrelativelyopentoSifthereexistsanopensetUCRnsuchthatUintersectS=V.Tomethisseemslikethesamedefinitionasthedefinitionofanopenset,andI'mstrugglingtocomeupwithadifference.Cansomeoneclarifywhatthedifferenceis?Thanksinadvance!3commentssharesavehidereport100%UpvotedLoginorsignuptoleaveacommentLogInSignUpSortby:best level1·4yr.ago·edited4yr.agoNewUserConsiderS=[0,2],V=(1,2]andgiveStherelativetopology.CertainlySandVarenotopensubsetsofR,but,VisrelativelyopeninSasV=(1,3)∩SHopethathelps.Edited:switchedVandStoalignwiththeOP.5ReplyShareReportSaveFollowlevel1·4yr.ago·edited4yr.agoThisistodowiththesubspacetopology.ThepointisthatifSisasubsetofRnyoucanendowitwithatopology(calledthesubspacetopology)wheretheopensetsofSarepreciselythesetsS∩UwhereUisopeninRn.ConvinceyourselfthatthisisindeedatopologyonS,thatis:EachsetinthesubspacetopologyisactuallyasubsetofSTheemptysetandSarebothopenFiniteintersectionsandarbitraryunionsofopensetsareopenOfcourse,subsetsofSthatareopenwithrespecttothesubspacetopology(orasyousay,relativelyopentoS)neednotbeopeninRn.Forexample,takingStobe[0,1],thenconsider[0,1/2)whichisnotopeninR,butitisrelativelyopentoS,becauseitistheintersectionofSandsay,(-1/2,1/2).3ReplyShareReportSaveFollowlevel1·4yr.agoB.S.FTR,use⊂tomakethesubset-ofoperatoronReddit:⊂Also,∩becomes∩,∈becomes∈,→becomes→,and&becomes&(escapingtheampersandinHTMLentityexamples).3ReplyShareReportSaveFollowr/learnmathPostallofyourmath-learningresourceshere.Questions,nomatterhowbasic,willbeanswered(tothebestabilityoftheonlinesubscribers).253kMembers281OnlineCreatedNov29,2009JoinToppostsfebruary8th2018Toppostsoffebruary,2018Topposts2018helpRedditcoinsRedditpremiumaboutcareerspressadvertiseblogTermsContentpolicyPrivacypolicyModpolicyRedditInc©2022.AllrightsreservedBacktoTop Advertisement
延伸文章資訊
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