Domain理论旨在为理论计算机科学中的计算理论和语义理论等提供数学模型。上世纪八十年代起，国内外研究人员用广义度量空间乃至更一般的结构取代偏序集，建立了量化domain理论。序和拓扑之间的相互联系是domain理论的精髓。量化domain的序结构性质方面研究成果颇丰，但其拓扑性质的研究却不令人满意，根本原因在于缺失与量化偏序理论相匹配的量化拓扑理论。从量化拓扑的角度研究量化domain是亟须解决的问题之一，这就是本项目的研究任务。注意到广义度量空间在量化domain理论中的重要地位，本项目重点研究广义度量空间的量化拓扑性质。Lowen引入的approach空间与广义度量空间之间的关系类似于偏序集与拓扑空间之间的关系，因此我们以approach空间为量化拓扑结构，研究广义度量空间上量化版本的Scott拓扑和Lawson拓扑，在此基础上研究量化domain的函数空间和闭子范畴。

Domain theory aims to provide mathematical models for computation theory and semantic theory in theoretical computer science. From 1980’s, replacing partially ordered sets by generalized metric space or some more general mathematical entities, researchers have established the theory of quantitative domains. The connection between order and topology is the essence of domain theory. The study of order theoretic properties of quantitative domains is now quite fruitful, but, the investigation of the topological properties is unsatisfactory. The reason is the lack of a theory of quantitative topology that is compatible with the theory of quantitative order. So, it is an urgent problem to investigate quantitative domains from the viewpoint of quantitative topology. This is the task of this project. Because of the importance of generalized metric spaces in quantitative domains, the primary aim of this project is to study quantitative topological properties of generalized metric spaces. Since the relationship between generalized metric spaces and approach spaces introduced by Lowen is completely analogous to that between partially ordered sets and topological spaces, approach spaces will be taken as quantitative topologies in this project. Based on this, we will study the quantitative Scott topology and quantitative Lawson topology of generalized metric spaces, and then investigate function spaces and closed categories of quantitative domains.