Images Of Paracompact And Locally Compact Spaces

A central question of the idea of Alexandroff is to establish the relationships between various topological spaces and metric spaces by means of various mapping. Since Arhangel 'skii published the famous paper "Mapping and spaces" in 1966, topologists pay close attention to closed images of various metric spaces, in particular are interested in the question after L. Foged obtained the nice characterization of closed images of metric space. In recent years, the closed images of locally compact metric spaces have been studied extensively. The following are some related results, which characterize the closed mapping or gain the decomposition theorems of locally compact metric spaces. The following are equivalent for a regular space X : (1) X is a closed image of a locally compact metric space; (2) X is a Frechet space with aσ-hereditarily closure-preserving k -network; (3) X is a Frechet space withσ-hereditarily closure-preserving compact k -network; (4) X is a closed image of a metric space, and each of its closed first-countable subspace is locally compact; (5) X is a Frechet space with a point-countable k -network, and each of its closed first-countable subspace is locally compact.In this paper, we obtained a new characterization of closed images of locally compact metric spaces by means of weak-compact k-networks.A study of images of paracompact locally compact spaces under certain compact-covering mapping is an important question in general topology. Recently, the characterizations for certain L -images and certain sequence-covering compact images of paracompact locally compact spaces have been studied.In this paper, we establish the characterizations of images of paracompact locally compact spaces under some sequence-covering CL - mappings.This paper was parted into four chapters.In the first chapter, we introduced the background knowledge of related.In the second chapter, we studied the characterization of closed images of locally compact metric spaces, the main result obtained in the second chapter is:A space X is a closed image of a locally compact metric space if and only if it is a Fre%c het space with a point-countable weak-compact k -network.In the third chapter, we studied the characterization of images of paracompact locally compact spaces under certain sequence-covering CL - mappings, the main results obtained in the third chapter are: (see Theorem 3.4.1, Theorem 3.4.2, Theorem 3.4.3, Theorem 3.4.5, Theorem 3.4.7, Theorem 3.4.8 ).In the forth chapter, firstly, we show that a regular space with a locally countable weak-base is g -metrizable. Secondly, we establish the relationships between space with a locally countable weak-base (resp. spaces with a locally countable weak-base consisting of N₀-subspaces) and metric spaces (resp. locally separable metric spaces) by means of compact-covering maps, 1-sequence-covering maps, compact maps,Ï€-maps and ss -maps, and show that all these characterizations are mutually equivalent. Thirdly, we show that 1-sequence-covering, quotient, ss -maps preserve spaces with a locally countable weak base.