EQUALITY OF THE ISBELL AND SCOTT TOPOLOGIES ON FUNCTION SPACES OF c-SPACES

In this paper, we mainly consider the question of when the Isbell and Scott topologies coincide on the set [X -> L] of all continuous mappings from a topological space X to a dcpo L with the pointwise order. The main results are: (1) If L is a weak sober dcpo which is bi-complete, then (i) that the Isbell and Scott topologies coincide on [X -> L] for all c-spaces X implies that L is a pointed L-dcpo; (ii) that the Isbell and Scott topologies coincide on [X -> L] for all irreducible c-spaces X implies that L is an L-dcpo. (2) Let L be a quasicontinuous UBC-domain and X a c-space. If L has a least element or X is connected, then the Isbell and Scott topologies coincide on [X -> L]. (3) Let L be a quasicontinuous UFL-domain and the topological space X = (sic)(i is an element of I) X-i, where every X, is an irreducible Scott c-space and I is a nonempty finite set. If L has a least element or I is a singleton, then the Isbell and Scott topologies coincide on [X -> L].