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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)

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