
点星网与度量空间的映像
Point-star Networks and Images of Metric Spaces
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P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P且cs覆盖列的点星网.
(2)X具有P且sn覆盖列的点星网.
(3)X是Cauchy sn对称空间且具有σ-P的cs网.
(4)X是Cauchy sn对称空间且具有σ-P的sn网.
(5)X是度量空间的序列覆盖、π且σ-P映像.
(6)X是度量空间的1序列覆盖、紧且σ-P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 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.
cs有限集族 / cs覆盖 / 点星网 / 序列覆盖映射 / &sigma / -cs映射 {{custom_keyword}} /
cs-finite families / cs-coverings / point-star networks / sequence-covering mappings / σ-cs-mappings {{custom_keyword}} /
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国家自然科学基金资助项目(11801254,11471153)
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