On weak bases
\textit{F. Siwiec} [Pac. J. Math. 52, 233-245 (1974; Zbl 0285.54022)] introduced \(g\)-metrizable spaces as ones with a \(\sigma\)-locally finite weak base. This notation is motivated by the classical Nagata-Smirnov metrization theorem. \textit{D. Burke, R. Engelking} and \textit{D. Lutzer} [Proc. Am. Math. Soc. 51, 483-488 (1975; Zbl 0307.54030)] proved that a topological space \(X\) is metrizable if and only if \(X\) has a \(\sigma\)-hereditarily closure-preserving base. Then \textit{Y. Tanaka} [ibid. 112, 283-290 (1991; Zbl 0770.54031)] asked whether a space with a \(\sigma\)-hereditarily closure-preserving weak base is \(g\)-metrizable. In this paper the author gives a positive answer to this question. On the other hand, following \textit{L. Foged's} result [Pac. J. Math. 98, 327-332 (1982; Zbl 0478.54025)], \textit{Y. Tanaka} [op. cit.] and \textit{S. Lin} [Chin. Ann. Math., Ser. A 13, 3, 403-409 (1992; Zbl 0770.54030)] (in Chinese) independently showed that a \(g\)-first countable space is \(g\)-metrizable if it has a \(\sigma\)-hereditarily closure-preserving \(k\)-network. Hence \textit{S. Lin} in [Generalized metric spaces and maps, (Beijing: China) Science Press, (1995; Zbl 0940.54002)] (in Chinese) asked whether every weak base of a topological space is a \(k\)-network. Although a point-countable weak base is a \(k\)-network and for a space each of whose point is a \(G_{\delta}\)-set every weak base is a \(k\)-network, here the author gives a negative example. A typical example of a non \(g\)-first countable space is the sequential fan \(S_{\omega}\), which plays an important role in the investigation of \(g\)-metrizable spaces. In the second part of the paper the author applies this property to the study of mapping theorems on weak bases. He shows that \(\sigma\)-locally countable (\(\sigma\)-compact-finite) weak bases are preserved by closed irreducible sequence-covering mappings. The author also discusses a product theorem of spaces having a point-countable (\(\sigma\)-locally finite, \(\sigma\)-locally countable, \(\sigma\)-compact-finite) weak bases.
scientific article; zbMATH DE number 2173256
scientific article