Spaces With κ-systems Of Particular Properties And Related Results

This paper is composed of two parts. In the first part, we obtain the relationships between spaces with particular point-countable k-systems and paracompact locally compact spaces. In the second part, we establish some general theorems of internal characterizations of some sequence-covering L-images and closed images of paracompact local Ψ-spaces(Ψ is the property that implies paracompactness, is preserved by maps and is hereditary with respect to closed subspaces). The work generalize some corresponding results of Jinjin Li, Shou Lin, Zhaowen Li, Kedian Li and so on.In the first part, we have the following results.Result 1 ( Theorem 2.1) For a space X, the following conditions are equivalent.(1) X has a k-system with the property σ-P;(2) X is the compact-covering quotient ∑-P-image of a paracompact locally compact space;(3) X is the quotient E-P-image of a paracompact locally compact space.Result 2 ( Theorem 2.2) Suppose f : X → Y is a mapping, where X is a paracompact locally compact space. The following conditions are equivalent.(1) f is a point-countable(resp., locally countable, compact-countable, compact-finite, point-finite) mapping;(2) f is an L-mapping(resp., an SL-mapping, a CL-mapping, a k-mapping, a compact mapping).In the second part, we have the following results.Result 3 ( Theorem 3.4) For a space X, the following are equivalent:(1) X is the compact-covering(resp., 1-sequence-covering, sequence-covering, sequentially quotient) countable-to-one image of a paracompact local Ψ-space;(2) X is the compact-covering(resp., 1-sequence-covering, sequence-covering, sequentially quotient) L-image of a paracompact local Ψ-space;(3) X has a point-countable cfp-cover(resp., sn-cover, cs-cover, cs*-cover) consisting of the subsets with the property Ψ.Result 4 ( Corollary 3.5) For a space X, the following are equivalent:(1) X is the 1-sequence-covering finite-to-one image of a paracompact local Ψ-space;(2) X is the 1-sequence-covering compact image of a paracompact local Ψ-space;(3) X is the sequence-covering finite-to-one image of a paracompact local Ψ-space;(4) X is the sequence-covering compact image of a paracompact local Ψ-space;...