Relatively Open and Closed Sets | Math Forums

If Y is closed relative to X and p is a border point relative to X, then p has a neighborhood entireiy in Y relative to Y, so Y relative to Y is ... Menu MathForums Home Forums Trending Forums Login Register Menu RelativelyOpenandClosedSets Threadstarter zylo Startdate Mar7,2017 Tags closed open sets Home Forums UniversityMath/HomeworkHelp Topology Z zylo Mar2015 1,720 126 NewJersey RelativelyOpenandClosedSets Iperiodicallycomebacktothistotryandclearitupinmymind.ThistimeIwasgoingtoproposeitasaquestionandintheprocessanswereditmyself. UsiingNotationofRudinChapt2: LetXbeametricspace.That'stoovaqueandambiquous.LetX=(R^n). RELATIVETOX: 1)XopenbecauseeverypointofXhasaneighborhoodentirelyinX. 2)XclosedbecauseeverylimitpointofXisinX. LetYbeaclosedoropensubsetofX. RELATIVETOY: 3)YopenbecauseeverypointofYhasaneighborhoodentirelyinY. 4)YclosedbecauseeverylimitpointofYisinY. IfYisclosedrelativetoXandpisaborderpointrelativetoX,thenphasaneighborhoodentireiyinYrelativetoY,soYrelativetoYisopen. IfYisopenrelativetoX,andpisaborderpointrelativetoX,pdoesn'texistrelativetoY.WithrespecttoY,pointsnotinYdon'texist. Example. LeyYbeinteriorofacirclewithorwithoutitsborder(closedoropeninX).Drawasolidlinedownthemiddle.WithrespecttoY,thelineandareatorightoflineareclosed,andtheareatoleftoflineisopen,regardlessofwhetherornotYisopenorclosedwithrespecttoX. SowhenRudinsaysletXbeametricspace,itdoesn'thavetobespecifiedasclosedoropenorneither.   M Maschke Aug2012 3,395 1,244 zylosaid: LetXbeametricspace.That'stoovaqueandambiquous.LetX=(R^n). RELATIVETOX: 1)XopenbecauseeverypointofXhasaneighborhoodentirelyinX. 2)XclosedbecauseeverylimitpointofXisinX. Clicktoexpand... Iamnothappywith(1)and(2).Ifanopensetisdefinedby(1),thenaclosedsetisdefinedasthecomplementofanopenset.Then(2)islatershowntobeapropertyofclosedsets.Sothere'saslightlogicmismatchbetween(1)and(2).Minorpointbutinthecontextofyourquestionitwouldbebettertobeverypreciseaboutwhatisanopensetandwhatisaclosedset. I'mnotsurewhatquestionyouareasking.Butbeingopenisarelativeproperty. Example:Intherealnumbers$\mathbbR$,theset$\mathbbR$isopen,becauseeverypointisinterior. Butasasubsetofthecomplexnumbers$\mathbbC$,$\mathbbR$isNOTopen,becauseeveryopenballaroundanypointof$\mathbbR$containsnon-realcomplexnumbers. Right?Right.Sowhetheraparticularsetisopenornotisafunctionofwhattopologicalspaceit'sconsideredpartofinthecontextofaparticulardiscussion. Nowhereisapuzzlerforyou.Themetricspace$\mathbbR$iscomplete.EveryCauchysequenceconverges. Nowusingthetangent/arctangentwehaveahomeomorphism(bijectioncontinuousinbothdirections)between$\mathbbR$andtheopenunitinterval$(0,1)$.But$(0,1)$isNOTcomplete,sincetheCauchysequence$(\frac{1}{n})$doesnotconverge.Whatdoesthistellus?   Lastedited:Mar7,2017 Z zylo Mar2015 1,720 126 NewJersey Maschkesaid: Nowhereisapuzzlerforyou.Themetricspace$\mathbbR$iscomplete.EveryCauchysequenceconverges. Nowusingthetangent/arctangentwehaveahomeomorphism(bijectioncontinuousinbothdirections)between$\mathbbR$andtheopenunitinterval$(0,1)$.But$(0,1)$isNOTcomplete,sincetheCauchysequence$(\frac{1}{n})$doesnotconverge.Whatdoesthistellus? Clicktoexpand... Nothing.   Z zylo Mar2015 1,720 126 NewJersey Maschkesaid: Nowhereisapuzzlerforyou.Themetricspace$\mathbbR$iscomplete.EveryCauchysequenceconverges. Nowusingthetangent/arctangentwehaveahomeomorphism(bijectioncontinuousinbothdirections)between$\mathbbR$andtheopenunitinterval$(0,1)$.But$(0,1)$isNOTcomplete,sincetheCauchysequence$(\frac{1}{n})$doesnotconverge.Whatdoesthistellus? Clicktoexpand... Sorry,tookmeawhiletofigureoutwhatyouweretalkingabout. Questionsofcompletenessareirrelevantinopenandunboundedsets. Irrelevanttothisthread-whyIaskeditinnewthread: http://mymathforum.com/topology/339421-map-0-1-r.html IassumeyoumeanttodiscreditmeinordertodiscreditOP,whichmeansitiscorrect,oryouwouldn'thavebothered.   M Maschke Aug2012 3,395 1,244 zylosaid: Iassumeyoumeanttodiscreditme Clicktoexpand... I'veneverbeenanythingotherthanpoliteandhelpfultoyou.Doyoufeelotherwise?Canyoulinkanyexampletothecontrary?   Z zylo Mar2015 1,720 126 NewJersey Maschkesaid: I'veneverbeenanythingotherthanpoliteandhelpfultoyou.Doyoufeelotherwise?Canyoulinkanyexampletothecontrary? Clicktoexpand... WhatistherelevanceofyourpuzzlerquestioninPost#2toOP?   M Maschke Aug2012 3,395 1,244 zylosaid: WhatistherelevanceofyourpuzzlerquestioninPost#2toOP? Clicktoexpand... Isayagain.IntheyearorsothatyouandIhaveparticipatedincommonthreadsonthissite,haveIeverbeenlessthanpoliteandhelpfultoyou?Canyoulinkanexampleofsame? Thepurposeofthequestionwastomakeyouthink.Evidentlyyouaren'tinterestedindoingthat.Ifyou'relookingformetoberudetoyouI'mhappytooblige.   Z zylo Mar2015 1,720 126 NewJersey Maschkesaid: Nowhereisapuzzlerforyou.Themetricspace$\mathbbR$iscomplete.EveryCauchysequenceconverges. Nowusingthetangent/arctangentwehaveahomeomorphism(bijectioncontinuousinbothdirections)between$\mathbbR$andtheopenunitinterval$(0,1)$.But$(0,1)$isNOTcomplete,sincetheCauchysequence$(\frac{1}{n})$doesnotconverge.Whatdoesthistellus? Clicktoexpand... Ihavegiventheanswerinposts5and10of: http://mymathforum.com/topology/339421-map-0-1-r.html Apparentlyyoustilldon'tgetit. ThereisasequenceinRwhoseinversemapconvergesto0,whichisnotin(0,1),whichisnotsurprisingsince0doesn'tmaptoR.SotheinversemapofeveryconvergentsequenceinRisconvergent. Youshouldgiveyoursources: http://math.stackexchange.com/questions/392934/do-uniformly-continuous-functions-map-complete-sets-to-complete-sets Bytheway,theOPwasaboutrelativelyopenandclosedsets.Ifyouhaveanyotherunrelatedquestions,pleaseposttheminyourownthread.   Lastedited:Mar8,2017 Z zylo Mar2015 1,720 126 NewJersey Maschkesaid: Nowhereisapuzzlerforyou.Themetricspace$\mathbbR$iscomplete.EveryCauchysequenceconverges. Nowusingthetangent/arctangentwehaveahomeomorphism(bijectioncontinuousinbothdirections)between$\mathbbR$andtheopenunitinterval$(0,1)$.But$(0,1)$isNOTcomplete,sincetheCauchysequence$(\frac{1}{n})$doesnotconverge.Whatdoesthistellus? Clicktoexpand... Letfbe1-1(bijective)andcontinuousbetween(0,1)andR. 1/nconvergestoalimit(whichisnotin(0,1)). f(1/n)convergestoalimit(whichisinRbecauseRiscomplete).-> 1/nconvergestoalimitin(0,1).Contradiction. fisnotbijective.   Z zylo Mar2015 1,720 126 NewJersey Maschkesaid: Nowhereisapuzzlerforyou.Themetricspace$\mathbbR$iscomplete.EveryCauchysequenceconverges. Nowusingthetangent/arctangentwehaveahomeomorphism(bijectioncontinuousinbothdirections)between$\mathbbR$andtheopenunitinterval$(0,1)$.But$(0,1)$isNOTcomplete,sincetheCauchysequence$(\frac{1}{n})$doesnotconverge.Whatdoesthistellus? Clicktoexpand... Thisisnotapuzzler. IfaboundedsubsetofRwerecompleteitwouldbeclosedandfwouldbeboundedandsocouldn'tmaptoR.Ithastobeincomplete.   LoginorRegister/Reply SimilarMathDiscussions MathForum Date Proofthatrelativelyprime NumberTheory Apr6,2020 $|H|$isrelativelyprimeto$[G:H]$ AbstractAlgebra Sep30,2017 Connectedness-RelativelyClose Topology Jun20,2016 AcademicGuidanceHowManyRelativelyEqualStatisticsCoursesDoUniversitiesHave? Academic/CareerGuidance Apr18,2016 Similarthreads Proofthatrelativelyprime $|H|$isrelativelyprimeto$[G:H]$ Connectedness-RelativelyClose AcademicGuidance HowManyRelativelyEqualStatisticsCoursesDoUniversitiesHave? Home Forums UniversityMath/HomeworkHelp Topology Top Bottom