Domain理论是图灵奖获得者Dana Scott提出的一门综合序结构、拓扑结构、范畴结构、逻辑等结构的数学与理论计算机科学分支，其目的是利用数学对象为函数式程序语言提供合适的指称语义。. 本项目拟在拓扑空间上研究Domain理论中的交连续性，完善拓扑空间上的交连续性理论，重点是研究交连续拓扑空间的禁止子空间刻画；并应用交连续空间理论刻画拟Polish空间范畴中的指数对象，最终对拟Polish空间范畴中的笛卡尔闭满子范畴进行分类，并考察它们关于概率幂函子的封闭性。. 通过本项目研究，我们旨在拟Polish空间范畴中寻找合适的笛卡尔闭满子范畴，并探讨它们作为概率高阶函数式程序语言的指称语义的可能性，为这类程序语言提供坚实的数学基础。

Domain Theory, initially introduced by the Turing Award recipient-Dana Scott, is a research filed at the interface of Mathematics and Theoretical Computer Science which studies order-theoretic, topological, categorical and logical structures, for the purpose of providing denotational semantics for functional programming languages by employing mathematical entities. ..We propose to investigate and enhance the theory of meet-continuity from Domain Theory on general topological spaces, with an emphasis on establishing a forbidden-subspace characterization for meet-continuous topological spaces. We will seek to apply our theory of meet-continuity to characterize exponential objects in the category of quasi-Polish spaces and locate its maximal Cartesian closed full subcategories. Finally, these Cartesian closed full subcategories will be sent to test whether they are stable under the probabilistic powerdomain functor...By conducting the proposed research, we aim to find proper Cartesian closed categories of quasi-Polish spaces and investigate the possibility that they provide denotational semantics for probabilistic higher-ordered functional programming languages, in hopes of establishing a solid mathematical foundation for programming languages of this type.