Domain理论在拓扑学中的应用

Domain theory is an important branch of the theoretic computer science. In this project, we mainly study several topological problems of domain theory, such as compactness of domains and function spaces, the Isbell topology.of function spaces, the powerdomains characterized by topology. Furthermore,.some problems of locale theory are also studied, such as the direct limits of.locales. The main results in this work are listed as follows: the equivalent.theorem of the categorys of Smyth power semilattices and continuous domains,.and adjoints between quasicontinuous domains and their subcategorys.established by upper powerdomains; RW-spaces and compactness of function spaces for L-domains are be characterized, which answers an open problem listed in.“Open problem in Topology” in the case of L-domains; compactness of L-domains.are characterized by the Isbell topology of function spaces; the direct limit.of compact normal (compact normal connected) frames is compact normal (compact normal connected).

Domain 理论是理论计算机科学的重要分支。本项目主要研究了Domain 理论中的拓扑学问题，如domain 及其函数空间的紧性，函数空间的Isbell 拓扑，由拓扑描述的幂domain 结构等。此外，我们还研究了与Domain 理论和拓扑学密切相关的locale理论中的问题，如locale 的逆极限等。主要结果有：通过上幂domain 建立了Smyth幂半格与连续domain 范畴的等价定理及拟连续domain 及其子范畴间的伴随关系；刻画了RW 空间及L-domain 函数空间的紧性，在L-domain 情形下回答了列于名著《拓扑学中的公开问题》与此相关的一个问题；利用函数空间上的Isbell 拓扑刻画了紧L-domain；通过简化locale 的direct 极限结构证明了紧正规（紧正规连通）Frame的direct 极限是紧正规（紧正规连通）Frame。