关于偏序集的正规完备化若干问题研究

旨在研究qc-space的完全正则性及紧性，沟通c-space、qc-space等空间之间的关系，建立若干相关范畴之间的联系；讨论偏序集上的收敛类的可拓扑化问题，借助偏序集上的收敛结构给出Domain理论中一些连续性的新刻画；深入研究QFS-domain关于幂构造的封闭性问题；从正规完备化不变性的角度，给出Domain理论中一些重要连续性和分离性的新推广，揭示更多的不变性和非不变性质，建立正规完备化、序结构、拓扑结构的若干新联结，发展一个用正规完备化研究Domain理论、拓扑结构和序结构的新途径和方法。

In this project we will investigate the complete regularity of qc-space and the connection between the category of qc-space and the category of other spaces; The convergent classes in posets will be studied and many notions of continuity in Domain theory will be characterized by convergence; We also will discuss whether the classes of QFS-domains are closed under powerdomain construction. From the point of view of the completion-invariant property, some important notions of continuity in Domain theory will be generalized and more invariant and non-invariant properties are investigated. A new approach based on normal completion to study Domain theory, topological structure and ordered structure will be developed.