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1、ElectronicNotesinTheoreticalComputerScience83(2004)URL:http://www.elsevier.nl/locate/entcs/volume83.html18pagesHowDoDomainsModelTopologies?PawelWaszkiewicz1InstituteofComputerScienceJagiellonianUniversityKrak´ow,PolandBooleCentreforResearchinInformaticsUniversityCollegeCork,IrelandAbstractInthis
2、briefstudyweexplicitlymatchthepropertiesofspacesmodelledbydomainswiththestructureoftheirmodels.Weclaimthateachpropertyofthemodelledtopologyiscoupledwithsomeconstructinthemodel.Examplesarepairs:(i)first-countability-strictlymonotonemap,(ii)developability-measurement,(iii)metriz-ability-partialmetr
3、ic,(iv)ultrametrizability-tree,(v)Choquet-completeness-dcpo,andmore.Bymakingthiscorrespondencepreciseandexplicitwerevealhowdomainsmodeltopologies.1IntroductionTheideathatpropertiesofcertaintopologicalspacescanbestudiedviaanappropriatepartiallyorderedsetthat“approximates”or“models”thespaceisprese
4、ntinearlyworkssuchasLacombe[29],Martin-L¨of[38],Scott[42],andhasbeendevelopedfurtherintheworkofWeihrauchandSchreiber[44]andKamimuraandTang[24].Sincethen,theconnectionbetweendomaintheoryand“classical”mathematicshasbeenexploitedinavarietyofapplicationsincluding:realnumbercomputation[15],integratio
5、n[17],[6],[10]anddiffer-entialcalculus[13],geometry[12],dynamicalsystems,fractalsandmeasuretheory[7],[8],andbasicquantummechanics[5].2Thereisacommonpatterninalloftheaboveresearch:oneidentifiesatopologyτontheobjectsofinterestX(usuallyitisametricspace),thendefinespartialapproximantsoftheobjectsoutoft
6、heresourcesavailableinthespace(usuallythesearecertaincompactorclosedsets)andapartialorder1Email:pqw@ii.uj.edu.pl2Mostoftheapplicationsaresurveyedin[9].c2004PublishedbyElsevierScienceB.V.WaszkiewiczPbetweenthem.Theconstructionmakesthemodelledspacehomeomor-phictothesubsetofmaximalelementsofPinthes
7、ubspaceScotttopology:hX,τi∼=hmaxP,σ
8、maxPi.Lastly,havingpreparedthesetup,onestudiesthephenomenainthemodelledspaceviathedomain-theoretictoolsavailableforthemodel.Itcomesasnosurprisethatthefundamentalquestionofwhichtopologicals
