; TeX output 2006.03.07:1404 C Ghtml: html: Gz html: html:4 :̂Dt G G cmr17Computably7tBasedLoscallyCompactSpaces /cXQ cmr12PraulTVaylor sMarcrh7,2006 U*t: cmbx9Abstract э )o cmr9ASD;`(Abstract;StoneDualit9y)isare-axiomatisationofgeneraltopAologyinwhichthetopAology onaspaceistreated,anotasaninnitarylattice,butasanexpAonen9tialob jectofthesame category5astheoriginalspace,}withanassoAciatedlam9bda-calculus.}Inthispaper,}thisis sho9wntobAeequiv|ralenttoanotionof+j cmti9computablebasisaMforloAcallycompactsoberspacesor loAcales,in9volvingafamilyofopensubspacesandaccompan9yingfamilyofcompactones.This generalises5Sm9yth'seectively^givendomainsand5Jung'sStrong^proximitylattices.>P9art5of thedataforabasisistheinclusionrelationofcompactsubspaceswithinopAenones,)whic9h is~form9ulatedinloAcaletheoryastheway-below~relationonacontinuouslattice.%Thenitary propAertiesҙofthisrelationarec9haracterisedhere,includingtheWilkerconditionforthecover of
Eacompactspaceb9ytwoopAenones.DThereallineisusedasarunningexample,KBbeing closelyrelatedtoScott'sdomain*ofintervals.ASDdoAesnotusethecategoryofsets,butthe fullsubAcategoryofo9vertdiscreteob jectspla9ysthisrole;itisanarithmeticuniversem(pretopos withlists).;4Inparticular,"&w9eusethissubAcategorytotranslatecomputablebasesforclassical spacesTin9toob jectsintheASDcalculus.@VConCtents &9. BasicTcorollariesohtml:32 html: 1.&LoAcallyTcompactsoberspaces b)html:2 html: