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Closed mapping theorems in $$k$$-spaces with point-countable $$k$$-networks. (English) Zbl 0832.54011
In the theory of $$k$$-spaces with point-countable $$k$$-networks, the author introduced the notion of “$$s$$-networks”. For a space $$X$$, a network $${\mathcal P}$$ for $$X$$ is an $$s$$-network, if for any non-closed subset $$A$$ of $$X$$, there exists a point $$x \in X$$ such that, for any open neighbourhood $$U$$ of $$x$$, some $$P \in {\mathcal P}$$ satisfies $$P \subset U$$, and $$P \cap A$$ is infinite.
Every point-countable $$s$$-network is a $$k$$-network, and every $$k$$-network for a $$k$$-space is an $$s$$-network. Also, for a closed map $$f : X \to Y$$, if $$X$$ has a point-countable $$s$$-network, then so does $$Y$$.
By means of $$s$$-networks, the author proved that any closed map $$f : X \to Y$$ is compact-covering when $$X$$ is a $$k$$-space with a point-countable $$k$$-network, and $$X$$ is regular. Also, he gave a decomposition theorem for a closed map $$f : X \to Y$$ under $$X$$ being a $$k$$-space with a point- countable $$k$$-network, which generalizes Lashnev’s decomposition theorem where $$X$$ is metric.
Reviewer: Y.Tanaka (Tokyo)

##### MSC:
 54B10 Product spaces in general topology 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
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