## Points of openness and closedness of some mappings.(English)Zbl 1322.54010

Let $$X$$ and $$Y$$ be topological spaces and $$f:X\to Y$$ be a continuous map. We say that $$f$$ is closed at $$y\in Y$$ if for every open neighbourhood $$W\subset X$$ of $$f^{-1}(y)$$ there is a neighbourhood $$V$$ of $$y$$ such that $$f^{-1}(V)\subset W$$. $$f$$ is open at $$x\in X$$ if it maps neighbourhoods of $$x$$ into neighbourhoods of $$f(x)$$. $$f$$ is open at $$y\in Y$$ if for each open set $$A\subset X$$, $$y\in f(A)$$ implies $$y\in Int f(A)$$. In the paper under review the authors study conditions on $$X$$ and $$Y$$ under which the set of $$y\in Y$$ at which $$f$$ is open, and the set of $$y\in Y$$ at which $$f$$ is closed are $$G_\delta$$ sets in $$Y$$. They generalize some results of S. Levi, R. Engelking and I. A. Vaĭnšteĭn. The main idea of these generalizations is to replace of the assumption that $$Y$$ is first countable by the weaker one, that $$Y$$ is a $$w$$-space. Recall that the notion of $$w$$-space has been introduced by G. Gruenhage in [General Topology Appl. 6, 339–352 (1976; Zbl 0327.54019)]. A typical result: Let $$X$$ be a completely metrizable space, $$Y$$ be a Hausdorff $$w$$-space, and let$$f:X\to Y$$ be a continuous map. Then the set of all points of $$Y$$ at which $$f$$ is closed is a $$G_\delta$$ set.

### MSC:

 54C10 Special maps on topological spaces (open, closed, perfect, etc.) 54C05 Continuous maps 54C60 Set-valued maps in general topology

Zbl 0327.54019
Full Text: