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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

Citations:

Zbl 0327.54019
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