On weak-open compact images of metric spaces

To find internal characterizations of certain images of metric spaces is one of central problems in general topology. Arhangel’skiı̆ [1] showed that a space is an open compact image of a metric space if and only if it has a development consisting of point-finite open covers, and some characterizations for certain quotient compact images of metric spaces are obtained in [3, 5, 8]. Recently, Xia [12] introduced the concept of weak-open mappings. By using it, certain g-first countable spaces are characterized as images of metric spaces under various weak-open mappings. Furthermore, Li and Lin in [4] proved that a space is g-metrizable if and only if it is a weak-open σ-image of a metric space. The purpose of this paper is to give some characterizations of weak-open compact images of metric spaces, which showed that a space is a weak-open compact image of a metric space if and only if it has a weak development consisting of point-finite cs-covers. In this paper, all spaces are Hausdorff, all mappings are continuous and surjective. N denotes the set of all natural numbers. τ(X) denotes the topology on a space X . For the usual product space ∏ i∈NXi, πi denotes the projection ∏ i∈NXi onto Xi. For a sequence {xn} in X , denote 〈xn〉 = {xn : n∈N}.