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Borel $\sigma$-ideal; Hurewicz test
\font\mm=cmbx10 at 12pt \def\boldSigma{\mm\char6{}} \def\boldPi{\mm\char5{}} We develop the theory of topological Hurewicz test pairs: a concept which allows us to distinguish the classes of the Borel hierarchy by Baire category in a suitable topology. As an application we show that for every ${\boldsymbol \Pi}^{0}_{\xi}$ and not ${\boldsymbol \Sigma}^{0}_{\xi}$ subset $P$ of a Polish space $X$ there is a $\sigma$-ideal $\Cal I\subseteq 2^{X}$ such that $P\notin \Cal I$ but for every ${\boldsymbol \Sigma}^{0}_{\xi}$ set $B\subseteq P$ there is a ${\boldsymbol \Pi}^{0}_{\xi}$ set $B'\subseteq P$ satisfying $B\subseteq B'\in \Cal I$. We also discuss several other results and problems related to ideal generation and Hurewicz test pairs.
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