Abstract
This paper is dedicated to Professor A.V. Arhangel'skiı̌'s 80th birthday. In this paper, we discuss some relations among point-regular covers, uniform covers and σ-point-finite covers of a topological space, express spaces with a point-regular special family as images of metric spaces under certain compact mappings, and prove that the following statements are equivalent for a subset A of a topological space X.(1)X has a point-regular cs-network at A for X.(2)X has a point-regular sn-network at A for X.(3)X has a uniform cs-network at A for X.(4)X has a uniform sn-network at A for X.(5)There are a metric space M and a mapping f:M→X satisfying the following conditions: for each x∈A, f−1(x) is compact in M; and there is a point z∈f−1(x) such that f(U) is a sequential neighborhood of x in X whenever U is a neighborhood of z in M. As some applications of these results, some characterizations of certain compact images of metric spaces are obtained.
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