函数空间的拓扑结构  

Topological Structures of Function Spaces

作　　者：杨寒彪 杨忠强[2] YANG Hanbiao;YANG Zhongqiang(School of Mathematics and Computational Science,Wuyi University Jiangmen 529020,Guangdong,China;Department of Mathematics,College of Science,Shantou University,Shantou 515063,Guangdong,China)

机构地区：[1]五邑大学数学与计算科学学院,广东江门529020 [2]汕头大学理学院数学系,广东汕头515063

出　　处：《汕头大学学报（自然科学版）》2021年第2期3-13,F0002,共12页Journal of Shantou University：Natural Science Edition

基　　金：广东省教育科学十三五规划资助项目(2018GXJK192).

摘　　要：函数空间的拓扑结构研究主要考虑下面的问题:假设有一族函数构成的集合,其上赋予了某种自然的拓扑,因此构成了拓扑空间,研究这个空间的拓扑性质,进而探讨这个空间是否同胚于一个经典的空间.本文总结了这个问题的一些经典结果和最新结果.例如,同胚于?詛2的拓扑空间,Hilbert方体Q-流形和?詛2-流形的拓扑特征,非拓扑完备的无限维流形的拓扑特征等.根据时间顺序介绍了函数空间拓扑结构的历史发展顺序,总结了函数空间的拓扑结构在各数学分支的应用情况.The topological structure of function space mainly considers the following problems:Suppose there is a set of functions,which is endowed with a certain natural topology,so it forms a topological space.The topological properties of this space are studied,and then whether this space is homeomorphic to a classical space is discussed.In this paper,some classic and recent results of this problem are summarized.For example,the following results are given:the topological spaces of being homeomorphic to Hilbert space proved by Anderson,the topological characteristics of Hilbert cube manifolds and Hilbert space manifolds proved by Toruńczyk,and the topological characteristics of non topologically complete infinite dimensional manifolds,etc.The historical development sequence of topological structure of functional space is introduced according to the time sequence,and the applications of those results in various mathematical branches are summarized.

关 键 词：无限维拓扑学 函数空间 拓扑结构 

分 类 号：O189.11[理学—数学]