; TeX output 2003.08.05:2223 C Ghtml: html: Gz html: html:4 dDt G G cmr17Sobser7tSpacesandConqtinuations /cXQ cmr12PraulTVaylorfj August5,2003 U*t: cmbx9Abstract
/ )o cmr9AtopAological,spaceissoberifithasexactlythepoin9tsthataredictatedbyitsopAensets. W:euexplaintheanalogywiththew9ayuinwhic9hcomputationalv|raluesaredeterminedbythe observ|rationsXthatcanbAemadeofthem.Anewdenitionofsobriet9yisformulatedinterms of 5lam9bAdacalculusandelementarycategorytheory:,:withnoreferencetolatticestructure, but,USforHtopAologicalspaces,thiscoincideswiththestandardlattice-theoreticdenition.The primitiv9e؝symbAolicandcategoricalstructuresareextendedtomaketheirtypAessober.fKF:or theLnaturaln9umbAers,YtheLadditionalstructurepro9videsdenitionbydescriptionandgeneral recursion.&W:eusethesamebasiccategoricalconstructionthatThielec9ke,FA;uhrmannandSelinger use`ztostudycon9tinuations,sDbut`zouremphasisiscompletelydieren9t:we`zconcentrateonthe fragmen9tJSoftheircalculusthat+j cmti9excludescomputationaleects,Wbutshowhowitnevertheless denesdnewdenotationalxv|ralues.Noristhis\denotationalseman9ticsofcontinuationsusing sobAerTspaces",thoughthatcouldeasilybederiv9ed.&Onthecon9trary:,thispapAerprovidestheunderlying,5" cmmi9-calculusonthebasisofwhich abstractStonedualit9ywillre-axiomatisegeneraltopAology:.Theleadingmodelofthenew axiomsTisthecategoryofloAcallycompactlocalesandcon9tinuousTmaps.7s1lK`y
cmr10Contents Ka6 `EnforcingTsobriet9yhtml:19 html: 1.ComputationalTv|ralues lhtml:1 html: