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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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