点星网与度量空间的映像

拓扑空间X的覆盖列{P_i}_(i∈N)被称为空间X的点星网,若x∈X,则{st(x,P_i)}_(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.

Point-star Networks and Images of Metric Spaces

A sequence {P_i}_i∈N of covers of a topological space X is called a pointstar network for X if the family {st(x, P_i)}_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 P of set families satisfies some suitable conditions: (1) X has a point-star network consisting of cs-coverings with property P. (2) X has a point-star network consisting of sn-coverings with property P. (3) X is a Cauchy sn-symmetric space with a σ-P cs-network. (4) X is a Cauchy sn-symmetric space with a σ-P sn-network. (5) X is a sequence-covering, π and σ-P-image of a metrizable space. (6) X is a 1-sequence-covering, compact and σ-P-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.