\(\omega\)-closed mappings. (English) Zbl 0574.54008

The concepts of \(\omega\)-closed set, \(\omega\)-closed mapping and \(P^*\)- spaces are defined and the following are the main results: (a) Let f be a continuous \(\omega\)-closed mapping of a space X onto a space Y such that \(f^{-1}(y)\) is Lindelöf for each Y’ in Y. Then X is Lindelöf if Y is so. (b) Let f be a continuous \(\omega\)-closed mapping of a regular space X onto a space Y. Then X is paracompact (strongly paracompact) if Y is paracompact (strongly paracompact) and for each y in Y, \(f^{-1}(y)\) is paracompact relative to X (Lindelöf). (c) Let X be a Lindelöf space and Y be a \(P^*\)-space, then the projection \(P: X\times Y\to Y\) is an \(\omega\)-closed mapping. Hence, \(X\times Y\) is Lindelöf (paracompact, strongly paracompact) if and only if Y is so.


54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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