Domain理论是数学与理论计算机科学交叉研究的重要领域，该理论在连续偏序集和拟连续domain等数学结构上的拓展是当前研究的热点。本项目拟结合domain理论、模糊集理论、拓扑和范畴论等相关理论，基于已有研究结果，从以下四个方面展开研究：(1) 借助D-完备化这一有力工具，研究连续偏序集范畴的笛卡尔闭的满子范畴；(2) 以形式球为工具，基于模糊度量空间的domain表示理论，探讨广义模糊度量空间的domain表示；(3) 借鉴函数空间连续的相关研究方法，讨论特定拓扑空间到拟连续domain上的函数空间拟连续和其上Isbell拓扑与Scott拓扑一致的条件；(4) 对偏序集的拓扑性质进行研究，将domain中基在拓扑下的稠密性结果推广到s2连续偏序集，将良滤dcpo上关于Scott拓扑的coherence性的等价刻画推广至非定向完备偏序集。本项目的研究将大大地丰富和完善domain理论。

Domain theory is a very important area between Mathematics and Theoretical Computer Sciences. One of the most important topics is to generalize its results to continuous posets and quasi-continuous domains. Based on the existing results on domain theory, the project will launch the research from the following four aspects by combining domain theory, fuzzy set theory, topology and category theory. (1) Using the tool of D-completion, studying the Cartesian closed full subcategories of continuous posets; (2) Based on the domain-theoretical approach to fuzzy metric spaces, discussing the domain representation of generalized fuzzy metric spaces by the associated formal balls; (3) By the use of the methods on continuity of functional spaces, investigating the possible conditions for a functional space from a special topology space to a quasi-continuous domain to be quasi-continuous and the possible conditions under which the Isbell topology and the Scott topology is equal; (4) Studying the topological properties of posets, generalizing the denses of basis via topology on domain to s2 continuous posets and generalizing the characterization of coherence under Scott topology on well-filtered dcpos to the posets which is not a dcpo. These works will enrich and perfect the domain theory.