Scott--Isbell Coincidence for Continuous Dcpos beyond Bicompleteness

Abstract

Lawson and Mislove posed the following problem in 1990 as Problem~535 in \emph{Open Problems in Topology}: for a core-compact space \(X\) and a dcpo \(P\) equipped with its Scott topology, under what conditions on \(P\) do the Isbell and Scott topologies on \(C(X,P)\) agree?It was proved that, for a nonempty bicomplete continuous dcpo \(P\), the Isbell and Scott topologies on \(C(X,P)\) coincide for every core-compact space \(X\) if and only if \(P\) is bounded complete; for every compact core-compact space \(X\) if and only if \(P\) is conditionally bounded complete; and for every RW-space \(X\) if and only if \(P\) is a pointed continuous \(L\)-domain.We remove the bicompleteness assumption from all three classifications by combining the forbidden-retract theorem of Jia, Jung and Li with a separation theorem for powers of downward well-ordered chains and suitable Alexandrov test spaces.