Characterizations of certain \(g\)-first countable spaces (Q2767350)

A continuous and onto map \(f:X\to Y\) is called weakly open if there are a weak base \({\mathcal B}=\bigcup_{y\in Y}{\mathcal B}_y\) for \(Y\) and an \(x_y\in f^{-1}(y)\) for each \(y\in Y\) such that whenever \(U_y\) is a neighborhood of \(x_y\) in \(X\) \(B_y\subset f(U_y)\) for some \(B_y\in{\mathcal B}_y\). Every weakly open map is a quotient map. In this paper certain \(g\)-first countable spaces are characterized as images of metric spaces under various weakly open mappings. For example, it is shown that a space \(Y\) has a point-countable weak base if and only if it is a weakly open \(s\)-image of a metric space. Finally, some relation between weakly open maps and 1-sequence-covering maps is discussed. Here a map \(f:X\to Y\) is called 1-sequence-covering if for each \(y\in Y\) there is \(x_y\in f^{-1}(y)\) such that whenever \(\{y_n\}\) is a sequence converging to \(y\) in \(Y\) there is a sequence \(\{x_n\}\) converging to \(x_y\) in \(X\) with each \(x_n\in f^{-1}(y_n)\).