Spaces Defined By Weak-bases And Related Results

The classified principle of mapping and spaces,that is,Alexandroff-Arhangel'skii's idea is to establish the intrinsic law of various categories of topological spaces by means of mappings.A lot of topologists who are following this idea,study the intrinsic characterizations for images and inverse images of metric spaces under various mappings.The concept of weak-base,an important concept in theory of generalized metric spaces,was introduced by A.V.Arhangel'skii.Various spaces defined by weak-bases belong to generalized metric spaces,in which g-metrizable spaces(i.e.spaces withσ-locally finite weak-bases)are further studied by some topologists in the world,such as F.Siwiec,Y.Tanaka,L.Foged,Lin Shou,Liu Chuan and so on,and many noticeable results have been obtained.The purpose of this paper is to study some other spaces defined by weak-bases,and the paper consists of the following five parts.The first part includes Chapter 1.The historical background and the recent development of the problems are reviewed and the main contents of the paper are outlined. At the same time,we introduce some preliminary knowledge.The second part consists of Chapter 2,Chapter 4,Chapter 5 and Chapter 6.We give the intrinsic characterizations for various spaces defined by weak-bases,i.e.,spaces with uniform weak-bases,spaces withσ-locally countable weak-bases,spaces with locally countable weak-bases and spaces withσ-compact finite weak-bases.We establish the relationships among them and metric spaces by means of weak-open mappings,msssmappings, ss-mappings,msk-mappings,some covering mappings,and so on.The concept of msk-mappings is firstly introduce by us.The third part includes Chapter 3.We give the characterizations of weak-openπ-images of metric spaces,and prove that a space is a weak openπ-image of metric space if and only if it is a g-developable space.The fourth part includes Chapter 7.A new mapping theorem on g-metrizable spaces is given.The fifth part includes Chapter 8.We define the concept of weak compact k-network,and give a new characterization for closd images of locally compact metric spaces.The main results obtained by us are as follows:Theorem 2.2.1 The following is equivalent for a space X:(1)X has a uniform weak-base; (2)X is a g-first countable space with a uniform sn-network;(3)X has a weak-development consisting of point-finite sn-covers;(4)X is a weak-open compact image of a metric space.Theorem 3.2.1 The following is equivalent for a space X:(1)X is a weak openπ-image of metric space;(2)X has a weak-development consisting of sn-covers;(3)X has a weak-development consisting of sn-covers;(4)X is a Cauchy space.(5)X is a g-developable space.Theorem 4.2.1 A space has aσ-locally countable weak-base if and only if it is a weak-open msss-image of a metric space.Theorem 4.2.2 For a space X,(1)(?)(2)(?)(3)below hold,(1)X has aσ-locally countable weak-base;(2)X is a g-first countable space with aσ-locally countable sn-network;(3)X is a g-first countable space with aσ-locally countable k-network.Theorem 5.2.7 The following is equivalent for a space X:(1)X has locally countable weak-base;(2)X is a compact-covering,quotient,compact and ss-image of a locally separable metric space;(3)X is a quotient,compact,and ss-image of a locally separable metric space;(4)X is a quotient,π-image and ss-image of a locally separable metric space;(5)X is a 1-seqence-covering,quotient and ss-image of a locally separable metric space.Theorem 6.2.3 For a space X,(1)(?)(2)(?)(3)(?)(4)below hold.(1)X has aσ-compact finite weak-base;(2)X is a k-space with theσ-compact finite sn-network;(3)X is a g-first countable space with aσ-compact finite sn-network;(4)X is a g-first countable space with aσ-compact finite k-network.Theorem 6.2.4 A space has aσ-compact finite weak-base,if and only if it is a 1-seqence-covering,quotient and msk-image of a metric space.Theorem 7.2.3 The following is equivalent for a space X:(1)X is a g-metrizable space.(2)X is a strong sequence-covering,quotient,πandσ-image of a metric space. (3)X is a sequence-covering,quotient,πandσ-image of a metric space.(4)X is a quotient,πandσ-image of a metric space.Theorem 8.2.5 A space is a closed-image of a locally compact metric space if and only if it is a Frechet space with a point-countable weak compact k-network.