Ordered Spaces, Metric Preimages, and Function Alg

arXiv:math/0512553v2[math.GN]21Mar2007OrderedSpaces,MetricPreimages,andFunctionAlgebras∗KennethKunen†‡February2,2008AbstractWeconsidertheComplexStone-WeierstrassProperty(CSWP),whichisthecomplexversionoftheStone-WeierstrassTheorem.IfXisacompactsubspaceofaproductofthreelinearlyorderedspaces,thenXhastheCSWPifandonlyifXhasnosubspacehomeomorphictotheCantorset.Inaddition,everyfinitepowerofthedoublearrowspacehastheCSWP.TheseresultsareprovedusingsomeresultsaboutthosecompactHausdorffspaceswhichhavescattered-to-onemapsontocompactmetricspaces.1IntroductionAlltopologiesdiscussedinthispaperareassumedtobeHausdorff.Asusual,asubsetofaspaceisperfectiffitisclosedandnon-emptyandhasnoisolatedpoints,soXisscatterediffXhasnoperfectsubsets.TheusualversionoftheStone-WeierstrassTheoreminvolvessubalgebrasofC(X,R),andistrueforallcompactX.IfonereplacestherealnumbersRbythecomplexnumbersC,the“theorem”istrueforsomeXandfalseforothers,soitbecomesapropertyofX:Definition1.1IfXiscompact,thenC(X)=C(X,C)isthealgebraofcontin-uouscomplex-valuedfunctionsonX,withtheusualsupremumnorm.A⊑C(X)meansthatAisasubalgebraofC(X)whichseparatespointsandcontainstheconstantfunctions.A⊑cC(X)meansthatA⊑C(X)andAisclosedinC(X).∗2000MathematicsSubjectClassification:Primary54C40,46J10.KeyWordsandPhrases:Orderedspace,functionalgebra.†UniversityofWisconsin,Madison,WI53706,U.S.A.,[e-mail]‡AuthorpartiallysupportedbyNSFGrantDMS-0456653.11INTRODUCTION2XhastheComplexStone-WeierstrassProperty(CSWP)iffeveryA⊑C(X)isdenseinC(X);equivalently,iffeveryA⊑cC(X)equalsC(X).TheCSWPiseasilyseentobetrueforfinitespaces.Thecomplexanalysisdevelopedinthe1800sshowsthattheCSWPisfalseformanycompactsubspacesoftheplane;forexample,itisfalsefortheunitcircleT;theclassiccounter-examplebeingthealgebraofcomplexpolynomialsP⊑C(T).TheseremarksaresubsumedbyresultsofW.Rudin[14,15]fromthe1950s:Theorem1.2LetXbeanycompactspace.1.IfXcontainsacopyoftheCantorset,thenXfailstheCSWP.2.IfXisscattered,thenXsatisfiestheCSWP.Ifacompactspaceismetrizable(equivalently,secondcountable),thenitcon-tainsaCantorsubsetiffitisnotscattered,soasRudinpointedout:Corollary1.3IfXiscompactmetric,thenXsatisfiestheCSWPiffXdoesnotcontainacopyoftheCantorset.OnemightconjecturethatthiscorollaryholdsforallcompactX,butthatwasrefutedin1960byHoffmanandSinger[9](seealso[4,8]);theirresultsimplythatanycompactumcontainingβNfailstheCSWP.However,thecorollarydoesholdforsomemore“reasonable”classesofspaces.Kunen[11]showedin2004:Theorem1.4IfXisacompactLOTS,thenXsatisfiestheCSWPiffXdoesnotcontainacopyoftheCantorset.Asusual,aLOTSisalinearlyorderedtopologicalspace.Ofcourse,the→ofthisresultisclearfromTheorem1.2;onlythe←wasnew.Thistheoremshowsthattherearesomenon-scatteredspaceswiththeCSWP,suchasthedoublearrowspaceofAlexandroffandUrysohn(seeDefinition2.1,or[1],p.76).Onecannowaskwhethertherearefurtherclassesof“reasonable”spacesforwhichresultssuchasCorollary1.3andTheorem1.4hold.Wedonotknowthebestpossibleresultalongthisline,butweshallproveinSection5:Theorem1.5IfXiscompactandX⊆L0×L1×L2,whereL0,L1,L2areLOTSes,thenXhastheCSWPiffXdoesnotcontainacopyoftheCantorset.1INTRODUCTION3Here,wemayassumethatL0,L1,L2arecompact(otherwise,replacethembytheprojectionsofX).ItisunknownwhethertheproductoftwospaceswiththeCSWPmustalsohavetheCSWP.Evenifthisturnsouttobetrue,Theorem1.5isnotimmediatefromTheorem1.4,sinceXisanarbitrarycompactsubsetoftheproduct,andL0,L1,L2mayfailtheCSWP(i.e.,haveCantorsubsets).Byaslightlydifferentargument,weshallshowinSection7:Theorem1.6IfListhedoublearrowspace,thenLnhastheCSWPforeveryfiniten.Theorems1.5and1.6areprovedusingsomeresultsfromSection3aboutspaceswhichhavescattered-to-onemapsontometricspaces.InTheorem1.6,thereisanaturalf:Ln։[0,1]nforwhichtheinverseofeachpointisscattered(andofsize2n).InTheorem1.5,theLjneednothaveanyscattered-to-onemapsontometricspaces,butastandardargumentusingmeasuresreducestheproofofTheorem1.5tothecasewheretheLjareseparable(seeSection4),inwhichcaseXmusthaveaneight-to-onemapontoacompactmetricspace.IfL0,L1,L2areseparableinTheorem1.5,thenXmustalsobefirstcountable,andhence“small”inthecardinalfunctionssense(seeJuh´asz[10]).However,wedonotbelievethatthereisanotionof“reasonable”involvingonlycardinalfunctions.In[6]itisshownthatinsomemodelsofsettheory,thereisacompactXwhichdoesnotcontainCantorsubsetsandwhichfailstheCSWP,suchthatXisbothhereditarilyseparableandhereditarilyLindel¨of(andhencealsofirstcountable).Inthesemodels,2ℵ0=ℵ1andthestandardcardinalfunctionsofourX(alleitherℵ0orℵ1)aretheleastpossibleamongnon-metriccompacta.Section2reviewssomeelementaryfactaboutLOTSes.Section6discussesthenotionofaremovablespacedefinedin[5];thisisastrengtheningoftheCSWPusedinSection7.Definition1.7LetKbeaclassofcompactspaces.Kisclosed-hereditaryiffeveryclosedsubspaceofaspaceinKisalsoinK.KislocaliffKisclosed-hereditaryandforeverycompactX:ifXiscoveredbyopensetswhoseclosureslieinK,thenX∈K.Classesofcompactawhichrestrictcardinalfunctions(firstcountable,secondcountable,countabletightness,etc.)areclearlylocal,whereastheclassofcom-pactawhicharehomeomorphictoaLOTSisclosed-hereditary,butnotlocal.I