Discussion On Continuity And Compactness Of Function Space

It is important and essential to research function space in Domain Achim Jung proved that [D → E] implied either D is Lawson compact or E is L - domain when D and E are DCPO with property m and least element. Liu Yingming and Liang jihua proved that for all core compact space X a DCPO D is continuous L - domain if and only if [X → D] is continuous L - domain. But we know little about continuity and compactness of function space [X → D] for D which is not a L - domain.In this paper, we consider a interesting example M1 which is nerther a B - domain nor a L - domain. We prove that [X →M1] is continuous DCPO for all stable space X, furthermore, I shell and Scott topology on [X→ M1] agree. For Lawson compact in function space,we give two examples to prove that not every function space from a stable space to M\ or to B - domain is Lawson compact, which showes that [X → L] is continuous Lawson compact for all stable space X implies L is a L - domain, where L is a continuous DCPO with least element.