$$\omega$$-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.

MSC:

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