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Nagata-Smirnov revisited: spaces with \(\sigma\)-WHCP bases. (English) Zbl 1123.54009
A collection \(\mathcal H\) of subsets of a topological space \(X\) is weakly hereditarily closure-preserving (WHCP) if, whenever a point \(p(H)\in H\) is chosen for each \(H\in\mathcal H\), the set \(\{p(H): H\in\mathcal H\}\) is a closed discrete subspace of \(X\). In 1975, Burke, Engelking and Lutzer proved that each \(k\)-space with a \(\sigma\)-WHCP base is metrizable, and gave a regular space with a \(\sigma\)-WHCP base which is not metrizable. In this paper some metrization theorems for spaces with a \(\sigma\)-WHCP base are discussed. It is shown that a separable space with a \(\sigma\)-WHCP base is metrizable.
Reviewer: Shou Lin (Fujian)

54E35 Metric spaces, metrizability
54D65 Separability of topological spaces
54C10 Special maps on topological spaces (open, closed, perfect, etc.)