An \(\aleph_0\)-weak base for a space \(X\) consists of an assignment \((x,n)\mapsto\mathcal{B}_x(n)\) of families of subsets such that each \(\mathcal{B}_x(n)\) is closed under finite intersections and has \(x\) in its intersection, and \(U\subseteq X\) is open iff for all \(x\in U\) and \(n\in\mathbb{N}\) there is \(B\in\mathcal{B}_x(n)\) with \(B\subseteq U\).
The author uses these structures to characterize various quotient images of metrizable spaces. E.g., a space has a \(\sigma\)-locally finite \(\aleph_0\)-weak base iff it is a quotient of a metrizable space by a countable-to-one \(\sigma\)-map, where \(f\) is a \(\sigma\)-map if there is a base \(\mathcal{B}\) for its domain such that \(\bigl\{f[B]:B\in\mathcal{B}\bigr\}\) is \(\sigma\)-locally finite. Also, a space with a \(\sigma\)-closure preserving \(\aleph_0\)-weak base is hereditarily metaLindelöf if it is regular and hereditarily paracompact if it is normal.