Borel complexity up to the equivalence (Q2291605)

The families \(\mathcal F\) of subsets of the hyperspace \(\mathcal K([0,1]^\omega)\) of compact subsets of \([0,1]^\omega\) are the main objects of the investigation. Two such families are equivalent (up to homeomorphism) if each element of one of them has its homeomorphic copy in the other one. Among the elements of the sets \([\mathcal F]\subset\mathcal K\) of families equivalent with \(\mathcal F\), families \(\mathcal B\) of the possibly lowest complexity are examined. It is recalled that for every analytic family \(\mathcal A\subset\mathcal K\) there is a Polish subspace \(\mathcal P\) of the hyperspace \(\mathcal K\) which is equivalent to \(\mathcal A\), see also [\textit{A. Bartoš} et al., Topology Appl. 266, Article ID 106836, 25 p. (2019; Zbl 1429.54042)]. Using compactifiability studied in the mentioned paper, it is proved that for every \(F_{\sigma}\) family in \(\mathcal K\) there is a closed equivalent family. Some more constructions are needed to get open families \(\mathcal O_n\subset\mathcal K\), \(n\in\omega\), such that the equivalence classes \([\mathcal O_n]\) are distinct and they are the unique classes of equivalence up to homeomorphism which contain an open family. Corresponding results on families of continua are also derived.