分段线性连续函数的下方图形拓扑结构  

The Topological Structure of Regions Below of Piecewise Linear Continuous Maps

作　　者：陈满春[1] 黄若婷 庄方敏 李婷爽 

机构地区：[1]韩山师范学院数学与统计学院,广东潮州521041

出　　处：《韩山师范学院学报》2015年第3期20-23,共4页Journal of Hanshan Normal University

基　　金：广东省大学生创新创业培训项目(项目编号:NO.1057813048)

摘　　要：设S*={1/n:n∈N+}为一收敛数列.用K表示从区间(0,1]到[0,1]且分段点之集为S*的分段线性连续函数全体.USC表示单位闭区间到自身的所有上半连续函数全体.对任意f∈USC,↓f表示f的下方图形,即↓f={(x,t)|x∈I,0tf(x)}.对任意USC的子集A,令↓A={↓f|f∈A},对↓USC赋予Hausdorff度量拓扑,并对K中的每个函数补充其在0点的函数值为其上极限使K变为USC的子集,记为L.将证明↓L同胚于s=(0,1)∞,其中s为希尔伯特方体Q=[0,1]∞的子空间.Let S^＊={1/n：n ∈ N＋} be a convergent sequence. Let K denote all Piecewise linear continuous functions from（0,1] to [0,1],whose set of section points is S＊. USC denotes the set of all upper semicontinuous maps from unit closed interval to itself. For every f ∈ USC,↓ f denotes the region below of f,that is↓ f ={（x,t）｜x ∈ I,0 t f（x）}. For any＂subset A of usc＂,let ↓ A ={↓ f ｜ f ∈ A}. This paper endowes ↓ USC with Hausdorff metric topology. For every function in K,we endow its value on 0 with its upper limit,then K turn to be another set,denoted by L. This paper proves ↓ L is homeomorphic to s =（0,1）^∞,where s is a subspace of Hilbert cube Q =[0,1]^∞.

关 键 词：分段线性函数 同胚 上半连续函数 HAUSDORFF度量 希尔伯特方体 

分 类 号：O189.11[理学—数学]