Ukrainian Mathematical Bulletin
Volume 3, Issue 4, 2006 pp. 501-527.
Topological Aspects of Hurewicz Tests for the Difference HierarchyAuthors: Tamas Matrai
Author institution: Tamas Matrai, Department of Mathematics and its Applications, Central European University, N'ador u. 6, 1051 Budapest, Hungary, matrait@renyi.hu
Summary: We generalize the Baire Category Theorem to the Borel and diffe\-ren\-ce hierarchies, i.e., if $\Gamma$ is any of the classes $\Sigma^{0}_{\xi}$, $\Pi^{0}_{\xi}$, $D_{\eta}(\Sigma^{0}_{\xi})$ or $\check{D}_{\eta}(\Sigma^{0}_{\xi})$ we find a representative set $P_{\Gamma} \in \Gamma$ and a Polish topology $\tau_{\Gamma}$ such that for every $A \in \check{\Gamma}$ from some assumption on the size of $A\cap P_{\Gamma}$ we can deduce that $A \setminus P_{\Gamma}$ is of second category in the topology $\tau_{\Gamma}$. This allows us to distinguish the levels of the Borel and difference hierarchies via Baire category. We also present some typical Baire Category Theorem-like applications of the results.
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