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э )o cmr9TherstpapAerpublishedon+j cmti9AÎbstractStoneDuality^(sho9wedthattheovertdiscrete>ob jects (those\admitting- cmsy99and=in9ternally)formapretopos,i.e.\acategorywithnitelimits,stable disjoin9t|coproAductsandstableeectivequotientsofequiv|ralencerelations.Usingan0 a msbm9N-indexed leastsxedpAoin9taxiom,.hereweshowthatthisfullsubAcategoryisanarithmeticuniverse,.hav- ing_vafreesemilattice
(\collection_vofKuratowski-nitesubsets")andafreemonoidK(\collection oflists ")onan9yovertdiscreteob ject.%Eachnitesubsetisrepresentedbyitspair( ކ/ӱ msam91,}Z
%) ofmodalPoperators,althoughatigh9tcorrespAondencewiththesedependsonastrongerScott- con9tinuityaxiom.diT:opAologically,suc9hsubsetsarebAothcompactandopen5andalsoin9volve proper openmaps.InMapplicationsofASD:thiscaneliminatelistsinfa9vourMofacontinuation- passingN