Discussion Of The Topological Coincidence On Function Spaces

Function space is an essential structure in Domain theory. The problem when Isbell and Scott topology agree on function spaces plays an important rolein the study of topological structure on function spaces. Liu and Liang[19] obtained a solution to the problem in 1996. For studying the compactness of function spaces, Kou and Luo[13] defined property RW. In this paper we discuss further that for RW-space, the conditions on L such that the Isbell and Scott topologies on function space [X→L] agree.It's proved that the following theorems:(1) Let L be a pointed continuous domain with property m. Then the Isbell and Scott topologies on [X→L] agree for all core compact spaces X if and only if L is a bounded complete dcpo.(2) Let L be a continuous L-domain with the least element O_L. Then the Isbell and Scott topologies on [X→L] agree for all RW-spaces X if and only if L is Lawson compact.