A note on the hyperspace of finite subsets

In this paper, we study the relation between a space X satisfying certain generalized metric properties and its hyperspace of finite subsets F(X) satisfying the same properties. We prove that if F(X) is a stric B0-space then so is X. However, there exists a stric B0-space X such that Fn(X) is not a stric B0-space for each n ≥ 2, hence F(X) is not a stric B0-space. Moreover, we prove that X is a P-space (resp., sequentially separable) if and only if so is F(X).

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Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).