On the Cantor-Bendixson derivative, resolvable ranks, and perfect set theorems of A. H. Stone
Let \(X\) be a compact metric space and let \(2^X\) be the space of compact subsets of \(X\) with the Hausdorff metric. A Borel derivative on \(2^X\) is a Borel map \(D:2^X\to 2^X\) such that \(D(K)\subset K\) for \(K\in 2^X\). For example, the Cantor-Bendixson derivative is a Borel derivative. The iterations of \(D\) are defined inductively as follows: \(D^0(K)=K\), \(D^{\alpha+1}(K)=D(D^\alpha(K))\), and \(D^\alpha(K)=\bigcap_{\beta<\alpha}D^\beta(K)\) for limit \(\alpha\). The rank \(\delta:2^X\to\omega_1\cup\{\infty\}\) determined by \(D\) assigns to a compact set \(K\) the minimal \(\xi\) with \(D^{\xi+1}(K)=\emptyset\), if such \(\xi\) exists, or \(\infty\) if \(D^\xi(K)\neq\emptyset\) for all \(\xi\). The main result of the paper says that for any analytic set \(A\) in \(2^X\) if for some \(Z\subset A\) the set \(\{\xi:Z\cap\delta^{-1}(\{\xi\})\neq\emptyset\}\) is stationary in \(\omega_1\), then for all but non-stationary many \(\xi<\omega_1\) each \(F_\sigma\) set containing \(A\cap\delta^{-1}(\{\xi\})\) intersects both \(Z\cap\bigcup_{\alpha<\xi}\delta^{-1}(\{\xi\})\) and \(Z\cap\bigcup_{\alpha>\xi}\delta^{-1}(\{\xi\})\). With a Borel derivative \(D:2^X\to 2^X\) a partial order \(\preccurlyeq\) on \(2^X\) is determined by \(L\preccurlyeq K\) iff \(L=D^\alpha(K)\) for some \(\alpha\). The second main result of the paper says that if an analytic set \(A\subseteq 2^X\) intersects stationary many layers \(\delta^{-1}(\{\xi\})\) then \(A\) contains an \(\preccurlyeq\)-antichain intersecting all but non-stationary many layers \(\delta^{-1}(\{\xi\})\) in a Cantor set. These results are presented in a more general framework of resolvable ranks introduced in the paper.
scientific article; zbMATH DE number 1310635