Open Sets | Brilliant Math & Science Wiki

Open sets are the fundamental building blocks of topology. In the familiar setting of a metric space, the open sets have a natural description, ... Brilliant Home Courses Today Signup Login Excelinmathandscience. LoginwithFacebook LoginwithGoogle Loginwithemail JoinusingFacebook JoinusingGoogle Joinusingemail Forgotpassword? Newuser? Signup Existinguser? Login SignupwithFacebook or Signupmanually Alreadyhaveanaccount? Loginhere. RelevantFor... Geometry > Topology PatrickCorn, PeterMacgregor, and JiminKhim contributed Opensetsarethefundamentalbuildingblocksoftopology.Inthefamiliarsettingofametricspace,theopensetshaveanaturaldescription,whichcanbethoughtofasageneralizationofanopenintervalontherealnumberline.Intuitively,anopensetisasetthatdoesnotcontainitsboundary,inthesamewaythattheendpointsofanintervalarenotcontainedintheinterval. Thestandarddefinitionofcontinuitycanberestatedquiteconciselyintermsofopensets,andtheeleganceofthisrestatementleadstoapowerfulgeneralizationofthisideatogeneraltopologicalspaces.Inthesameway,manyotherdefinitionsoftopologicalconceptsareformulatedingeneralintermsofopensets. Thecomplementofanopensetisaclosedset.Manytopologicalpropertiesrelatedtoopensetscanberestatedintermsofclosedsetsaswell. Contents FormalDefinition Properties Continuity PropertiesDefinedusingOpenSets FirstStepsinPoint-setTopology References Inallbutthelastsectionofthiswiki,thesettingwillbeageneralmetricspace(X,d).(X,d).(X,d).ThosereaderswhoarenotcompletelycomfortablewithabstractmetricspacesmaythinkofXXXasbeingRn,{\mathbbR}^n,Rn,wheren=2n=2n=2or333forconcreteness,andthedistancefunctiond(x,y)d(x,y)d(x,y)asbeingthestandardEuclideandistancebetweentwopoints. Anopensetinametricspace(X,d)(X,d)(X,d)isasubsetUUUofXXXwiththefollowingproperty:foranyx∈U,x\inU,x∈U,thereisarealnumberϵ>0\epsilon>0ϵ>0suchthatanypointinXXXthatisadistance0\epsilon>0ϵ>0suchthatB(x,ϵ)B(x,\epsilon)B(x,ϵ)iscompletelycontainedinU.U.U. SomereferencesuseBϵ(x)B_{\epsilon}(x)Bϵ​(x)insteadofB(x,ϵ).B(x,\epsilon).B(x,ϵ).[1] Sotheintuitionisthatanopensetisasetforwhichanypointinthesethasasmall"halo"arounditthatiscompletelycontainedintheset.Theideaisthatthishalofailstoexistpreciselywhenthepointliesontheboundaryoftheset,sotheconditionthatUUUisopenisthesameassayingthatitdoesn'tcontainanyofitsboundarypoints.Withthecorrectdefinitionofboundary,thisintuitionbecomesatheorem. TheboundaryofasetSSSinsideametricspaceXXXisthesetofpointsssssuchthatforanyϵ>0,\epsilon>0,ϵ>0,B(s,ϵ)B(s,\epsilon)B(s,ϵ)containsatleastonepointinSSSandatleastonepointnotinS.S.S. AsubsetUUUofametricspaceisopenifandonlyifitdoesnotcontainanyofitsboundarypoints. ItisclearthatanopensetUUUcannotcontainanyofitsboundarypointssincethehaloconditionwouldnotapplytothosepoints.Ontheotherhand,ifasetUUUdoesn'tcontainanyofitsboundarypoints,thatisenoughtoshowthatitisopen:foreverypointx∈U,x\inU,x∈U,sincexxxisnotaboundarypoint,thatimpliesthatthereissomeballaroundxxxthatiseithercontainedinUUUorcontainedinthecomplementofU.U.U.ButeveryballaroundxxxcontainsatleastonepointinU,U,U,namelyxxxitself,soitmustbetheformer,andxxxhasahaloinsideU.U.U.□_\square□​ Trivialopensets:TheemptysetandtheentiresetXXXarebothopen.Thisisastraightforwardconsequenceofthedefinition. Unionandintersection:Theunionofanarbitrarycollectionofopensetsisopen.Theintersectionoffinitelymanyopensetsisopen. Toseethefirststatement,considerthehaloaroundapointintheunion.SinceanyxxxintheunionisinoneoftheopensetsU,U,U,ithasaB(x,ϵ)B(x,\epsilon)B(x,ϵ)arounditcontainedinU,U,U,sothatballiscontainedintheunionaswell.□_\square□​ Thesecondstatementisprovedinthebelowexercise. Theintersectionofinfinitelymanysetsisnotnecessarilydefined Theproofworks;thestatementistrue BBBmightnotbecontainedinsideUUU UUUmightbeempty BBBmightnotbeaballaroundxxx LetUαU_{\alpha}Uα​(α∈A)(\alpha\inA)(α∈A)beacollectionofopensetsinR2.{\mathbbR}^2.R2.IfAAAisfinite,thentheintersectionU=⋂αUαU=\bigcap\limits_\alphaU_{\alpha}U=α⋂​Uα​isalsoanopenset.Hereisaproof: Supposex∈U.x\inU.x∈U.Foreachα∈A,\alpha\inA,α∈A,letBαB_{\alpha}Bα​beaballofsomepositiveradiusaroundxxxwhichiscontainedentirelyinsideUα.U_{\alpha}.Uα​.ThentheintersectionoftheBαB_{\alpha}Bα​isaballBBBaroundxxxwhichiscontainedentirelyinsidetheintersection,sotheintersectionisopen. (((HereaballaroundxxxisasetB(x,r)B(x,r)B(x,r)(rrrapositiverealnumber)consistingofallpointsyyysuchthat∣x−y∣0\delta>0δ>0suchthatwhenever∣x−a∣