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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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##### References:
 [1] Borsuk K.; Ulam S., On symmetric products of topological spaces, Bull. Amer. Math. Soc. 37 (1931), no. 12, 875-882 [2] Gabriyelyan S. S.; Kakol J., On $$\mathfrak{P}$$-spaces and related concepts, Topology Appl. 191 (2015), 178-198 [3] Good C.; Macías S., Symmetric products of generalized metric spaces, Topology Appl. 206 (2016), 93-114 [4] Peng L.-X.; Sun Y., A study on symmetric products of generalized metric spaces, Topology Appl. 231 (2017), 411-429 [5] Tang Z.; Lin S.; Lin F., Symmetric products and closed finite-to-one mappings, Topology Appl. 234 (2018), 26-45
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