Spaces with Σ-locally Countable Weak-bases

In this paper, spaces with σ-locally countable weak-bases are characterized as the weakly open msss-images of metric spaces (or g-first countable spaces with σ-locally countable cs-networks). To find the internal characterizations of certain images of metric spaces is an interesting research topic on general topology. Recently, S. Xia introduced the concept of weakly open mappings, by using it, certain g-first countable spaces are characterized as images of metric spaces under various weakly open mappings. The present paper establish the relationships spaces with σ-locally countable weakbases and metric spaces by means of weakly pen mappings and msss-mappings, and give a characterization of spaces with σ-locally countable weak-bases. In this paper, all spaces are regular and T1, all mappings are continuous and surjective. N denotes the set of all natural numbers. ω denotes N ∪ {0}. For a family P of subsets of a space X and a mapping f : X → Y , denote f(P) = {f(P ) : P ∈ P}. For the usual product space ∏ i∈N Xi, pi denotes the projection from ∏ i∈N Xi onto Xi. Definition 1. Let P = ∪{Px : x ∈ X} be a family of subsets of a space X satisfying that for each x ∈ X , (1) Px is a network of x in X , (2) If U, V ∈ Px, then W ⊂ U ∩ V for some W ∈ Px. P is called a weak-base for X [1] if G ⊂ X is open in X if and only if for each x ∈ G, there exists P ∈ Px such that P ⊂ G. A space X is called g-first countable if X has a weak-base P such that each Px is countable. A space X is called a g-metrizable space if X has a σ-locally finite weak-base. Definition 2. Let P be a cover of a space X . 2000 Mathematics Subject Classification: 54E99, 54C10.