Some sequence-covering locally countable images of locally separable metric spaces (Q2732438)

Let \(X\) be a space. A collection \({\mathcal P}\) of subsets of a space \(X\) is called a cs*-network of \(X\), if whenever \(\{x_n\}\) is a sequence converging to a point \(x\in X\) and \(U\) is a neighborhood of \(x\) in \(X\), then for some \(P\in{\mathcal P}\) and some subsequence \(\{x_{n_k}\}\) of \(\{x_n\}\) such that \(\{x_{n_k}: k\in \mathbb{N}\}\cup \{x\}\subset P\subset U\). Let \(f:X\to Y\) be a continuous and onto map. \(f\) is a sequence-covering map, if each convergent sequence of \(Y\) is the image of some compact subset of X\(;\) \(f\) is a locally countable map, if for any star-countable base \({\mathcal B}\) for \(X\), \(f({\mathcal B})\) is locally countable in \textit{Y. Y. Tanaka} and \textit{S. Xia} [Quest. Answers Gen. Topology 14, No. 2, 217-231 (1996; Zbl 0858.54030)] discussed some properties of the images of separable metric spaces. In this paper the authors obtain some new characterizations of the images of locally separable metric spaces. For examples, a space \(X\) is a sequence-covering and locally countable image of a locally separable metric space if and only if \(X\) has a locally countable cs*-network.NEWLINENEWLINENEWLINEIt is still an open problem: What is a nice characterization for a quotient \(s\)-image of a locally separable metric space?