Non-Hausdorff拓扑和Domain理论中若干问题研究

从序、拓扑、范畴的交叉角度，研究non-Hausdorff拓扑和Domain理论中的一些基本问题和基本结构：(1) Scott稳定紧dcpo范畴SCD的cartesian闭性和概率幂domain构造的闭性；(2) d-空间范畴Top_d中具有sober性，cartesian闭性和幂构造闭性的（极大）满子范畴；(3) 定向空间范畴DTop的幂构造闭性；(4) 既约空间范畴IRTop的余反射性，cartesian闭性和幂构造闭性；(5) 涉及sober性、well-filtered性、Rudin空间、WD空间、强d-空间等的多个公开问题；(6) 建立一个讨论sober空间、well-filtered空间、d-空间及相关空间的统一框架，发展一个统一的理论和处理方法。本项目旨在解决这些关键问题，进一步丰富non-Hausdorff拓扑和Domain理论，促进序与拓扑的交叉融合。

In this project, from the point of view of the intersection of ordered structure, topology and category, we will investigate some fundamental problems and basic structures in non-Hausdorff topology and domain theory as follows: (1) the cartesian closedness and closedness under probabilistic powerdomain construction of the category SCD of Scott stable compact dcpos; (2) (maximal) cartesian closed and probabilistic powerdomain construction closed subcategories with sober objects of the category Top_d of d-spaces; (3) cartesian closedness and probabilistic powerdomain construction closedness of the category DTop of directed spaces; (4) coreflective property, cartesian closedness and probabilistic powerdomain construction closedness of the category IRTop of irreducible determined spaces; (5) several open problems related to sobriety, well-filteredness, Rudin spaces, well-filtered determined spaces and strong d-spaces; and (6) providing a uniform approach to d-spaces, sober spaces and well-filtered spaces, and develop a general framework and method for dealing with all these spaces. By solving these key problems, we would like to enrich non-Hausdorff topology and domain theory, and promote the intersection and integration between order theory and topology.