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.