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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