Borel complexity up to the equivalence

We say that two classes of topological spaces are equivalent if each member of one class has a homeomorphic copy in the other class and vice versa. Usually when the Borel complexity of a class of metrizable compacta is considered, the class is realized as the subset of the hyperspace $\mathcal{K}([0, 1]^ω)$ containing all homeomorphic copies of members of the given class. We are rather interested in the lowest possible complexity among all equivalent realizations of the given class in the hyperspace. We recall that to every analytic subset of $\mathcal{K}([0, 1]^ω)$ there exist an equivalent $G_δ$ subset. Then we show that up to the equivalence open subsets of the hyperspace $\mathcal{K}([0, 1]^ω)$ correspond to countably many classes of metrizable compacta. Finally we use the structure of open subsets up to equivalence to prove that to every $F_σ$ subset of $\mathcal{K}([0, 1]^ω)$ there exists an equivalent closed subset.

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