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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.

##### MSC:
 54E40 Special maps on metric spaces 54D55 Sequential spaces
##### Keywords:
$$g$$-metrizable; $$g$$-second countable; weak base; sn-network
Zbl 0285.54022