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Some properties of subsets, almost closed mappings and paracompactness. (English) Zbl 1006.54026

A subset \(A\) of a topological space \(X\) is \(\alpha\)-Hausdorff iff for any two points \(a\in A\), \(b\in X\smallsetminus A\), there are disjoint open sets \(U\) and \(V\) containing \(a\) and \(b\) respectively. Several properties of \(\alpha\)-Hausdorff subsets, almost closed mappings and closed graphs are studied. Some of the results are as follows.
If \(E\) is an \(\alpha\)-Hausdorff retract of a space \(X\), then \(E\) is a closed set in \(X\). If \(f:X\to Y\) is an almost closed mapping of a space \(X\) into a space \(Y\) such that for each \(y\in f(X)\) \(f^{-1}(y)\) is an \(\alpha\)-Hausdorff \(\alpha\)-nearly paracompact subset with respect to \(X\smallsetminus f^{-1}(y)\), then \(f\) has a closed graph. If \(f:X\to Y\) is an almost closed mapping of a space \(X\) into a compact space \(Y\) such that the family \(\{f^{-1}(y)\mid y\in f(X)\}\) consists of \(\alpha\)-Hausdorff subsets which are mutually \(\alpha\)-nearly paracompact, then \(f\) is continuous.

MSC:

54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
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