On π-s-images of metric spaces

In 1966, Michael [11] introduced the concept of compact-covering maps. Since many important kinds of maps are compact-covering, such as closed maps on paracompact spaces, much work has been done to seek the characterizations of metric spaces under various compact-covering maps, for example, compact-covering (open) s-maps, pseudosequence-covering (quotient) s-maps, sequence-covering (quotient) s-maps, and compact-covering (quotient) s-maps, see [3, 9, 12, 15, 16]. π-map is another important map which was introduced by Ponomarev [13] in 1960 and correspondingly, many spaces, including developable spaces, weak Cauchy spaces, g-developable spaces, and semimetrizable spaces, were characterized as the images of metric spaces under certain quotient π-maps, see [1, 4, 6, 7]. The purpose of this paper is to establish the characterizations of metric spaces under compact-covering (resp., pseudo-sequence-covering, sequence-covering) π-s-maps by means of cfp-covers (resp., sfp-covers, cs-covers) and σ-strong networks. In this paper, all spaces are Hausdroff, and all maps are continuous and surjective. N denotes the set of all natural numbers. ω denotesN∪{0}. τ(X) denotes a topology on X . For a collection of subsets of a space X and a map f : X → Y , denote { f (P) : P ∈ } by f ( ). For the usual product space ∏ i∈NXi, πi denotes the projective ∏ i∈NXi onto Xi. For a sequence {xn} in X , denote 〈xn〉 = {xn : n∈N}. Definition 1.1. Let f : X → Y be a map. (1) f is called a compact-covering map [11] if each compact subset of Y is the image of some compact subset of X . (2) f is called a sequence-covering map [14] if whenever {yn} is a convergent sequence in Y , then there exists a convergent sequence {xn} in X such that each xn ∈ f −1(yn).