A mapping theorem on ℵ-spaces

Abstract. In this paper we give a mapping theorems on ℵ-spaces by means of strong compact-covering mappings and σ-mappings. As an application, we get characterizations of quo-tient (pseudo-open) σ-images of metric spaces. ℵ-spaces form one class of generalized metric spaces, and play an important role in metrization theory. The concept of σ-mappings was introduced by S.Lin in [5], and by using it, σ-spaces are characterized as σ-images of metric spaces. In [17], g-metrizable spaces are characterized as weak-open σ images of metric spaces. The purpose of this paper is to establish the relationships between metric spaces and ℵ-spaces by means of strong compact-covering mappings and σ-mappings, and get characterizations of quotient (pseudo-open) σ-images of metric spaces. In this paper all spaces are regular and T1, all mappings are continuous and surjective. N denotes the set of natural numbers, ω denotes N∪{0}. For a collection P of subsets of a space X and a mapping f: X → Y, denote f(P) = {f(P) : P ∈ P}. For two families A and B of subsets of X, denote A∧B = {A∩B: A ∈ A and B ∈ B}. Definition 1. Let P be a cover of a space X. (1) P is a k-network for X [9] if for each compact subset K of X and its open neighborhood V, there exists a finite subcollection P ′ of P such that K ⊂ ∪P ′ ⊂ V. (2) P is called a cs-network for X if for each x ∈ X, its open neighborhood V and a sequence {xn} converging to x, there exists P ∈ P such that {xn: n ≥ m} ∪ {x} ⊂ P ⊂ V for some m ∈ N. (3) P is called a cs∗-network for X if for each x ∈ X, its open neighborhood V and a sequence {xn} converging to x, there exist P ∈ P and a subsequence {xnk} of {xn} such that {xnk: k ∈ N} ∪ {x} ⊂ P ⊂ V. A space X is called an ℵ-space if X has a σ-locally finite k-network