Some Problems On Function Spaces On Domains

With the development of computer science, the research about the fundamental of computer science is paid attention to more and more. It have been the common area that is interested by the researchers of the computer science and the mathematicians'. Domain theory, which appeared in the early seventies of twenty century, is the just one. The introduction of Domain theory is originated from two different kinds of background: one is the study of functional language in the theoretical computer science, the other is the study of pure mathematics. From the two different kinds of background, the same object is discovered, which stimulates the study of this area. In 1980, Scott and other co-authors wrote the famous book about continuous lattice [15]. The complementary version was published in 2003[16], containing the results of the latest twenty years. After about 30 years, the study of this area has been making progress, making the close relationship with logic, category, topology, lattice theory, locale theory , topology on lattices and other mathematical areas and branches [2,4,7,12,13,63,76,89 et.al.], meanwhile, some interesting problems are presented. Edited by Jan van Mill and George M.Reed, published in 1990, 《 open problems in topology 》 has the special chapter which many problems of this area is presented in. There are many problems which refer to the function spaces on Domains.In the paper, some problems on function spaces on Domains are considered. Let X be a topological space, and L a DCPO with Scott topology. Denote the set of continuous functions from X to L as [X → L]. With the pointwise order, [X → L] is still a DCPO. In the first part of the paper, we prove that, for a continuous B-domain L, the function space [X → L] is continuous for each core compact and coherent space X. Generally, the discussion on the Lawson compactness of the function space [X → L] is difficult. Many knownresults was obtained under the condition that L is an L-Domain. For the case that L is not L-Domain, since the continuity of the function spaces is not clear, the difficulty still exists. Basing on the previous work about the continuity of function spaces, we can consider this problem for the case that L is a continuous B-Domain. One typical example is given and shows that, when L is not L-Domain, the compactness of the function spaces is hard to be discussed further. It also shows from other side that L-Domain is the most proper category to discuss compactness and then some doubts are cleared.Isbell and Scott topologies are the two important topologies on function spaces. In [64], Mislove and Lawson posed the following problems:Let X be a core compact space and L a DCPO equipped with the Scott topology. Under what condition on L do the Isbell and Scott topologies on [X —? L] agree?Academician Liu Yingming and Professor Liang Jihua solved this problems by means of the following theorem.Theorem[54] Let L be a continuous L-Domain with least element O^. Then the Isebell and Scott topologies on [X—> L] agree for all core compact spaces X if and only if L is a bounded complete DCPO.We show that the conditions of this theorem can be weakened and that their result is improved by means of the following statement.Theorem Let L be a continuous DCPO which is bicomplete. 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 with least element.If L is not L-Domain, when do the Isbell and Scott topologies agree on function spaces? We prove that, if X is a core compact and coherent space and L a continuous L-Domain with least element, then the Isbell and Scott topologies agree on functions space [X —? L] agree. In order to study the compactness of[X__? L], Kou and Luo [31] introduced the notion of RW-spaces. It is provedin this paper that, if L is a bicomplete continuous DCPO, then the Isbell and Scott topologies on [X —> L] agree for all RW-spaces X if and only if L is a L-Domain with least element. Moreover, it is obtained that, if X is a core compact and locally connected space, then the Isbell and Scott topologies agree on functions space [X —? L] for all continuous L-Domain L with least element if and only if X is an RW-space. Particularly, if X is a continuous DCPO, then Isbell and Scott topologies agree on function spaces [X —-+■ L] for all continuous L-Domain L with least element iff X is an RW-space. For the case that L is not L-Domain, it is proved that, if X is a core compact and coherent space and L is a continuous B-Domain, then the Isbell and Scott topologies agree on function spaces [X —> L}.If the topological space X has Domain hull, then X can embed into the maximal points Max(￡>) of continuous Domain D. Then the compact subsets of X correspond to the maximal points of convex power domain CD. Conversely, if each maximal points of convex power domain is generated by the compact subset of topological space X? On the set of the compact subsets of topological space, there are two topologies: one is Vietoris topologies; as the maximal points of convex power domain, the other is inherited from the Scott topology on convex power domain. A natural question is asked that if these two topologies agree? Professor Liang Jihua and doctor Kou Hui discussed this problem [51].The second part of the paper is the continue job of [51]. It is discussed that when the compact subset Com(Max(D)) of the maximal points Max(￡>) of continuous Domain D and the maximal points Max(CD) of convex power Domain CD are one-to-one correspondence. Particularly, it is proved that for upper space UX of locally compact T2 space X, Com(Max(￡/X))(i.e. the compact subset of X) and maximal points Max(C(UX)) of the convex power Domain C(UX) are one-to-one. In this case Lawson topology on upper space UX agrees with the Vietoris topology on Com(Max(UX)), which homeomorphic to Max(C (U X)) withthe Scott topology inherited from the C(UX).