Coincidence of the Isbell and Scott topologies on domain function spaces

Let [ X → L ] be the set of all continuous mappings from a topological space X to a domain L with the pointwise order. Let Is [ X → L ] , σ [ X → L ] be the Isbell and Scott topologies on [ X → L ] respectively. In this paper, the question of when the Isbell and Scott topologies coincide on [ X → L ] is considered. Main results are:(1) s a bicomplete domain, then (i) Is [ X → L ] = σ [ X → L ] for all core compact spaces X if and only if L is bounded complete; (ii) Is [ X → L ] = σ [ X → L ] for all core compact and compact spaces X if and only if L is conditionally bounded complete. be a dcpo consisting of a least element and a decreasing sequence with two lower bounds, then Is [ X → L ] = σ [ X → L ] for all core compact spaces X with a countable base. be a well rooted Lawson compact domain and L an FL-domain, then Is [ X → L ] = σ [ X → L ] .