Geodesic
Covering $\Sigma^0_\xi$-generated ideals by $\Pi^0_\xi$ sets
Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 2, pp. 245-268.

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\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.
Classification : 03E15, 54H05
Keywords: Borel $\sigma$-ideal; Hurewicz test 
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     title = {Covering $\Sigma^0_\xi$-generated ideals by $\Pi^0_\xi$ sets},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {245--268},
     publisher = {mathdoc},
     volume = {48},
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     zbl = {1199.54189},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2007__48_2_a6/}
}
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Mátrai, Tamás. Covering $\Sigma^0_\xi$-generated ideals by $\Pi^0_\xi$ sets. Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 2, pp. 245-268. http://geodesic.mathdoc.fr/item/CMUC_2007__48_2_a6/