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Notes on g-metrizable spaces. (English) Zbl 1085.54019
In this paper all spaces are regular and \(T_1\); all mappings are continuous and onto. A space is called \(g\)-metrizable if it has a \(\sigma\)-locally finite weak base, see F. Siwiec [Pac. J. Math. 52, 233-245 (1974; Zbl 0285.54022)]. The author studies spaces with a \(\sigma\)-HCP (resp., \(\sigma\)-wHCP) weak base. (Here “wHCP” means “weakly hereditarily closure preserving”.) In particular the following results are obtained: A space \(X\) is \(g\)-metrizable if and only if \(X\) has a \(\sigma\)-HCP weak base and has countable tightness. Under CH, a separable space with a \(\sigma\)-wHCP weak base is \(g\)-second countable, and thus \(g\)-metrizable; the author does not know whether CH can be omitted in this result. If \(X\) is a \(g\)-metrizable space and every perfect image of \(X\) has a \(\sigma\)-wHCP weak base, then \(X\) is metrizable. There is an \(\aleph_0\)-space that contains no copy of the sequential fan \(S_\omega,\) but is not sn-first countable.

54E40 Special maps on metric spaces
54D55 Sequential spaces
Zbl 0285.54022