Continuous Domains and its Relevant Problems
|School||Shaanxi Normal University|
|Keywords||Continuous domain Consistently connected continuous domain Maximal point space Z-connected continuous poset Dual equivalence|
Domain theory has always played an essential role in computer science and topology. Until now, there have been several definitions of domain, such as continuous domains, Z-continuous posets, generalized continuous posets. In this paper,we mainly disscuss continuous domains and its some relevant problems. On one hand, we investigate it in classical domain theory. Some properties of continuous domains which are similar to those of continuous lattices are obtained. The equalities of continuous domains, the relation of continuous domains with waybelow auxiliary order ,and some topological properties of the maximal point space of continuous domains are gotten. All these results embody the interior character of continuous domains. On the other hand, we study continuous domains in general aspects. The conceps of consistently connected continuous domains and Z-connected set system are introduced. Then, some good results about category are given.These conclusions supply some new cues for the study of continuous domains.Chapter one defines some notions and gives some theorems about continuous domains and category theory we will use in the paper.In chapter two, the notions of pseudodirected minimal set, pseudocontinuous domains and approximating auxiliary order are introduced firstly, and then some equivalent characterizations of pseudcontinuous domains and continuous domains are given respectively, and the equalities of continuous domains are obtained, and we proved a pseudocontinuous domain is a continuous domain if and only if it is a general continuous domain .In the end, by using the waybelow relation, a characteristic theorem of continuous domains is given.In chapter three, firstly, the concepts of connected sets, consistently connected sets and consistently connected continuous domains are introduced, and then it is proved that a consistently connected complete poset is a consistently connected continuous domain if and only if its principal ideals are all consistently connected continuous domains. Secondly, by using of the properties of the lattice of closed set, it is obtained that a consistently connected complete poset is a consistently connected continuous domain if and only if its closed set lattice is a complete completely distributive lattice and it has one-step closure property. In the end, the definition of connected completions is given and it is showed that the category of connected continuous domain is a full reflective subcategory of consistently connectedincontinuous domain.In chapter four, by using the equivalent characterizations of the maximal point spaces, we obtain that subspaces, disjoint sums, cartensian products, inverse limits of inverse sequence, and countable locally compact hausdorff spaces are all maximal point spaces. In particular, we give that second countable locally compact sober spaces are maximal point spaces. In the end, the domain hull of second countable locally compact hausdorff spaces is given.In chapter five, we mainly prove that the category of Z-connected continuous posets is dually equivalent to a full subcategory of the category of complete completely distributive lattices. Some other properties of Z-connected continuous posets are also investigated.