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Zbl 0692.54017
Yun, Ziqiu
On a problem of J. Nagata.
(English)
[J] Commentat. Math. Univ. Carol. 30, No.4, 811-815 (1989). ISSN 0010-2628

A function g: ${\bbfN}\times X\to 2\sp X$ is called a g-function if g(n,x) is an open neighborhood of x for all $n\in {\bbfN}$ and $x\in X$. A g-function g is said to be decreasing if $g(n+1,x)\subseteq g(n,x)$ for all $n\in {\bbfN}$ and $x\in X$. Let $g\sp 1(n,x)=g(n,x)$ and let $g\sp{i+1}(n,x)=\cup \{g(n,y):$ $y\in g\sp i(n,x)\}$ for $i\in {\bbfN}.$ \par The following problem was posed by {\it J. Nagata} [ibid. 29, 715-722 (1988; Zbl 0686.54019), problem after Theorem 9]: Is a $T\sb 1$ space X metrizable if X has a g-function which satisfies the following conditions: (1) If $x\in g\sp 2(n,x\sb n)$ for each $n\in {\bbfN}$, then $x\sb n\to x$; (2) For all $n\in {\bbfN}$ and $Y\subseteq X$, $CIY\subseteq \cup \{g\sp 2(n,y):$ $y\in Y\}?$ \par We answer this problem negatively and discuss some related problems.
MSC 2000:
*54E35 Metric spaces, metrizability

Citations: Zbl 0686.54019

Cited in: Zbl 0976.54029

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