Some Properties Of Countably Approximating Posets And Lattices Of Countably Scott-closed Sets

Domain theory is one of the important contents in theoretical computer science, and it appeared in order to lay mathematical foundations of semantics for computer languages. The researcher both in theoretical computer science and in mathematics have been interested in domain theory. One of important aspects of domain theory is to carry the theory of continuous lattices to that of posets as much as possible. Countably approximating posets are proved to be a successful generalization of continuous domains.In this thesis, we will give further studies on countably approximating posets. The main con-tents of the thesis consist of two parts. Firstly, we discuss some topological properties of countably approximating posets and the properties relative to continuous mappings in countably approxi-mating posets, and we give two characterizations of countably approximating posets. Secondly, we discuss the properties of the lattices of Scott-closed subsets. In particularly, we introduce the concept of countably C-continuous posets and we prove that the lattices of Scott-closed subsets in arbitrary posets is countably C-continuous. In addition, we discuss some properties of countably C-algebraic lattices.