Abstract: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.
Keywords: Borel $\sigma $-ideal, Hurewicz test
AMS Subject Classification: Primary 54H05; Secondary 03E15