点可数覆盖与序列覆盖映射——献给宁德师范高等专科学校

Point-Countable Covers And Sequence-Covering Mappings

映射与覆盖的方法是研究一般拓扑学的基本工具 .作为对可度量性与紧性一般化而形成的广义度量理论与覆盖性质理论中的许多问题涉及到对确定的点可数覆盖的研究 .与点可数覆盖相关的广义度量问题的探讨导致k网理论与度量空间的映射理论的发展 .本文在综述了广义度量空间理论在 90年代的主要研究课题及国内外学者的重要贡献后 ,分两个部分阐述了作者 (及其合作者 )近 3年在空间与映射方面的一些工作 .第一部分 (本文第二、三章 )讨论点可数覆盖、点有限覆盖列与度量空间的s映象、紧映象之间的关系 .第二部分 (本文第四章 )讨论著作《广义度量空间与映射》中的正则分离性条件及几个有失误的论证 .本文的第一部分围绕度量空间的几类序列覆盖映象中的一些问题开展研究 ,引入了序列网、点星网、sn覆盖和so覆盖等概念 ,利用了k网、紧有限分解网、cs网和sn网等集族性质 .主要的结果是证明了弱第一可数性与S2 ,Sω 之间的精巧关系 ,建立了度量空间的序列商映象、紧覆盖映象、序列覆盖映象与 1序列覆盖映象的特征 .其作用在于充实了序列覆盖映射的理论 ,深化了Arhangel’skii,Michael,Nogura ,Shibakov ,Svetlichny ,Velichko ,Tanaka等的一些定理 ,尤其是肯定地回答了下述问题 .(1)Arhangel’skii的问题[2 0 8] :

The method of mappings and covers is a basic tool to study general topology.The theory of generalized metric properties and covering properties stems from a generalization of metrizability and compactness,in which a great many problems involve a research for certain point countable covers.A deliberation on questions related to point countable covers leads to a development of the theories of k networks and mappings of metric spaces in generalized metric spaces.This paper summarizes the essential tasks and the important contribution to the theory of generalized metric spaces in the past ten years,and in two parts expounds some results obtained by author and cooperators concerning spaces and mappings in the past three years.In the first part,which is contained in the second and the third chapters,we discuss the relations among point countable covers,the sequences of point finite covers,and s images and compact images of metric spaces.In the second part,which is contained in the fourth chapter,we consider a regular separated axiom and some gaps in proofs with respect to the book “Generalized Metric Spaces and Mappings”. In the first part of this paper,we centre on several questions with sequence covering images of metric spaces.The concepts of sequential networks,point star networks,sn covers and so covers are introduced.The properties of families about k networks,cfp networks,cs networks and sn networks are made use of.The main results are that shows some exquisite relations among a weak first countability,S 2 and S ω,and that establishes certain characterizations of images of metric spaces under sequential quotient mappings,compact covering mappings,sequence covering mappings and 1 sequence covering mappings.Their role is to replenish the theory of sequence coverings mappings,and to deepen some theorems obtained by Arhangel’skii,Michael,Nogura,Shibakov,Svetlichny,Velichko,Tanaka and others.In particular,the following questions are answered affirmatively, (1) Arhangel’skii’s question [208] : If X is a sequential space with a weigh κ,then is X a quotient image of a metric space with a weigh κ? (2) Ikeda Liu Tanaka’s question [89] : For a sequential space X with a point regular cs network,characterize X by means of a nice image of a metric space. (3) Liu Tanaka’s question [149,218] : Let X be a k space with a point countable k network.If X contains no closed copy of S ω,and no S 2,then does X have a point countable base? (4) Liu Tanaka’s question [147] : Let X be a k space with a σ point finite k network.If X contains no closed copy of S ω,then is X a gf countable space? (5) Tanaka’s question [218] : Let X be a sequential space which contains no closed copy of S ω.If X has a point countable cs network,then does X have a point countable weak base? And,the following questions are answered partially, (6) Michael Nagami’s question [158] : If X is a quotient and s image of a metric space,then is X a quotient,compact covering and s image of a metric space? (7) Velichko’s question [231] : Find a Φ property such that a space X is a quotient and s image of a metric and Φ space if and only if X is a Φ space which is a quotient and s image of a metric space. In the second part of this paper,we concentrate our attention on a regular separated axiom and a few gaps in proofs of “Generalized Metric Spaces and Mappings”.Buhagiar’s idea studying fibrewise general topology is absorbed.The main results are that obtains several generalized metric theorems in T 2 spaces,and that opens up a research for subparacompactness.For example,it is proved that strong Σ * spaces are subparacompact,and that subparacompactness is invariant under perfect pre images,and some interesting examples are constructed which shows that (1) a σ closed discrete space without any G δ * diagonal,(2) a space with a locally countable and σ discrete k network without any point countable cs * network,and (3) a developable space without any p s