保持底空间度量的Fell拓扑的相容度量  

Metrics of the Fell topology that preserve the metric of the underlying space

作　　者：杨忠强 卫国 Yang Zhongqiang;Wei Guo(School of Mathematics and Statistics,Minnan Normal University,Zhangzhou 363000,China;Department of Mathematics and Computer Science,College of Arts and Science,University of North Carolina at Pembroke,Pembroke North Carolina 28372,USA)

机构地区：[1]闽南师范大学数学与统计学院,福建漳州363000 [2]北卡罗莱纳大学彭布罗克分校文理学院数学与计算机系,美国北卡罗莱纳彭布罗克28372

出　　处：《纯粹数学与应用数学》2021年第4期394-414,共21页Pure and Applied Mathematics

基　　金：国家自然科学基金(11971287)。

摘　　要：对于任意的局部紧可分(完备)度量空间(X,d),令Cld(X)为赋予了Fell拓扑的X上的非空闭集族.证明存在Cld(X)上一个相容(完备)度量D使得其保持X上的度量d.对于n维度欧式空间R^(n),可以构建这样的度量D.不幸的是,这个具体构造的度量D不是完备的.所以,寻找一个满足上面条件的完备度量是一个有意义的问题.而如果限制在R^(n)的非空凸闭集族Conv(R^(n)),则可以重新具体构造一个相容的具体的完备度量D_(Conv)使得其保持R^(n)上的度量.For any locally compact separable(complete)metric space(X,d),let Cld(X)denote the space of all non-empty closed subsets of X equipped with the Fell topology.It is proved that there exists a compatible(complete)metric D on Cld(X)that preserves the metric d of X.For the n-dimensional Euclidean space R^(n),such a metric D is explicitly constructed,but the constructed D on Cld(R^(n))is not complete.Hence,seeking a complete metric on Cld(R^(n))is an interesting problem.On the other hand,for the space Conv(R^(n))of all non-empty closed convex subsets of R^(n)equipped with the Fell topology,an explicit complete compatible metric D_(Conv)that preserves the usual metric of R^(n)has been constructed in this paper.

关 键 词：Fell拓扑的度量 等距扩张 绝对扩张 收缩 Alexandroff紧化 

分 类 号：O189.1[理学—数学]