点星网与度量空间的映像  

Point-star Networks and Images of Metric Spaces

作　　者：林寿 黄燕晖 张静 Shou LIN;Yan Hui HUANG;Jing ZHANG(School of Mathematics and Physics,Ningde Normal University,Ningde 352100,P.R.China;School of Mathematics and Statistics,Minnan Normal University Zhangzhou 363000,P.R.China)

机构地区：[1]宁德师范学院数理学院,宁德352100 [2]闽南师范大学数学与统计学院,漳州363000

出　　处：《数学学报（中文版）》2019年第6期865-878,共14页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(11801254,11471153)

摘　　要：拓扑空间X的覆盖列{Pi}i∈N被称为空间X的点星网,若x∈X,则{st(x,Pi)}i∈N是x在X中的网.本文刻画具有cs有限cs覆盖列的点星网的空间,并将其表示为度量空间在确定映射下的像.在假设集族性质β满足适当的条件下,证明对拓扑空间X下述条件相互等价:(1) X具有β且cs覆盖列的点星网.(2)X具有β且sn覆盖列的点星网.(3)X是Cauchy sn对称空间且具有σ-β的cs网.(4) X是Cauchy sn对称空间且具有σ-β的sn网.(5) X是度量空间的序列覆盖、π且σ-β映像.(6) X是度量空间的1序列覆盖、紧且σ-β映像.这些工作以局部有限集族与点有限集族为特例,拓展了从基到cs网的研究,丰富了映射与空间的相互分类思想.A sequence {Pi}i∈N of covers of a topological space X is called a pointstar network for X if the family {st(x,Pi)}i∈N is a network at x in X for each x∈X.The main purpose of this paper is to characterize the spaces which has a point-star network consisting of cs-finite cs-coverings and express their as certain images of metric spaces.It is proved that the following are equivalent for a topological space X when the property β of set families satisfies some suitable conditions:(1) X has a point-star network consisting of cs-coverings with property β.(2) X has a point-star network consisting of sn-coverings with property β.(3) X is a Cauchy sn-symmetric space with a σ-β cs-network.(4) X is a Cauchy sn-symmetric space with a σ-β sn-network.(5) X is a sequence-covering,π and σ-β-image of a metrizable space.(6) X is a 1-sequence-covering,compact and σ-β-image of a metrizable space.The above and some related work contain the study of locally finite or point-finite families as a special case,expand the research from bases to cs-networks,and enrich the idea of mutual classification on mappings and spaces.

关 键 词：cs有限集族 cs覆盖 点星网 序列覆盖映射 σ-cs映射 

分 类 号：O189.1[理学—数学]