Complements of sets in abstract Borelian hierarchies (Q803128)

Main result: For each \(1<\alpha <\omega_ 1\), there is a compact family \({\mathcal E}\) of subsets of a certain set E contained in [0,1]\(\cup [2,3]\) and a set \(A\in H_ 2({\mathcal E})\) such that \(E\setminus A\in H_{\alpha}({\mathcal E})\setminus \cup \{H_{\xi}({\mathcal E})| \quad \xi <\alpha \}.\) Definitions. Compact family means: if \({\mathcal F}\subset {\mathcal E}\) is such that \(\cap {\mathcal F}\) is empty then some finitely many members of \({\mathcal F}\) have empty intersection. \(H_ 0({\mathcal E})\) \(=\) set of all countable intersections of members of \({\mathcal E}\); \(H_ 1({\mathcal E})\) \(=\) set of all countable unions of members of \(H_ 0({\mathcal E})\); \(H_{\alpha}({\mathcal E})\) \(=\) set of all countable intersections of members of \(\cup \{H_{\beta}({\mathcal E})|\beta <\alpha \}\) if \(\alpha\) is even or limit ordinal, and \(H_{\alpha}({\mathcal E})\) \(=\) set of all countable unions of members of \(\cup \{H_{\beta}({\mathcal E})|\beta <\alpha \}\) if \(\alpha\) is odd or a successor ordinal.