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We study the relation between a space satisfying certain generalized metric properties and its -fold symmetric product satisfying the same properties. We prove that has a --property -network if and only if so does . Moreover, if is regular then has a --property -network if and only if so does . By these results, we obtain that is strict -space (strict -space) if and only if so is .
We study some generalized metric properties on the hyperspace of finite subsets of a space endowed with the Vietoris topology. We prove that has a point-star network consisting of (countable) -covers if and only if so does . Moreover, has a sequence of -covers with property which is a point-star network if and only if so does , where is one of the following properties: point-finite, point-countable, compact-finite, compact-countable, locally finite, locally countable. On the other...
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