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On the \(n\)-fold symmetric product of a space with a \(\sigma\)-\((P)\)-property \(cn\)-network (\(ck\)-network). (English) Zbl 07286004
Summary: We study the relation between a space \(X\) satisfying certain generalized metric properties and its \(n\)-fold symmetric product \(\mathcal{F}_n(X)\) satisfying the same properties. We prove that \(X\) has a \(\sigma\)-\((P)\)-property \(cn\)-network if and only if so does \(\mathcal{F}_n(X)\). Moreover, if \(X\) is regular then \(X\) has a \(\sigma\)-\((P)\)-property \(ck\)-network if and only if so does \(\mathcal{F}_n(X)\). By these results, we obtain that \(X\) is strict \(\sigma\)-space (strict \(\aleph\)-space) if and only if so is \(\mathcal{F}_n(X)\).
MSC:
54B20 Hyperspaces in general topology
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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