Spaces with \(\sigma\)-locally countable weak bases (Q2744267)

A mapping \(f:X\to Y\) is called an msss-mapping if there exists a product space \(\prod_{i\in\mathbb{N}}X_i\) satisfying (1) \(X\) is a subspace of \(\prod_{i \in\mathbb{N}} X_i\); (2) for each \(y\in Y\) there is a sequence \(\{V_i\}\) of open neighborhoods of \(y\) in \(Y\) such that each \(p_if^{-1} (V_i)\) is a separable subspace of the metric space \(X_i\). In this paper by msss-mappings, the relations between metric spaces and spaces with a \(\sigma\)-locally countable cs-network or spaces with \(\sigma\)-locally countable weak bases are established. For example, a Hausdorff space \(X\) has a \(\sigma\)-locally countable cs-network if and only if \(X\) is a sequence-covering msss-image of a metric space.