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Zbl 0661.54028
Yun, Ziquiu
On a question about metrizable spaces. (English)
[J] Quest. Answers Gen. Topology 6, No.2, 181-183 (1988). ISSN 0918-4732

The following question was posed by {\it J. Nagata} [this journal 4, 129- 139 (1987; Zbl 0616.54028)] question 5 and the remarks after it: Does the class of $T\sb 1$ spaces which have g-functions satisfying the following conditions (1), (2) and (3) coincide with the class of metrizable spaces? (1) If $\{x\sb n\}\to p\in X$ and $x\sb n\in g(n,y\sb n)$, then $\{y\sb n\}\to p$. (2) If $y\in g(n,x)$, then g(n,y)$\subseteq g(n,x)$. (3) For any $A\subseteq X$, Cl(A)$\subseteq \cup \{g(n,x):$ $x\in A\}.$ \par We prove that having g-functions which satisfy conditions (1) and (3) is a characterization of metrizable spaces, while the class of spaces which have g-functions satisfying the conditions (1), (2) and (3) coincides with the class of strongly zero-dimensional metrizable spaces. The former result gives a generalization of Nagata's theorem 1, and the latter one answers the above-mentioned question negatively.

MSC 2000:: *54E35 Metric spaces, metrizability
54E20 Stratifiable spaces, etc.

Keywords: g-functions; strongly zero-dimensional metrizable spaces

Citations: Zbl 0616.54028

Cited in: Zbl 0713.54029

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