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.
@article{CMUC_2007__48_2_a6,
author = {M\'atrai, Tam\'as},
title = {Covering $\Sigma^0_\xi$-generated ideals by $\Pi^0_\xi$ sets},
journal = {Commentationes Mathematicae Universitatis Carolinae},
pages = {245--268},
publisher = {mathdoc},
volume = {48},
number = {2},
year = {2007},
mrnumber = {2338093},
zbl = {1199.54189},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMUC_2007__48_2_a6/}
}
TY - JOUR AU - Mátrai, Tamás TI - Covering $\Sigma^0_\xi$-generated ideals by $\Pi^0_\xi$ sets JO - Commentationes Mathematicae Universitatis Carolinae PY - 2007 SP - 245 EP - 268 VL - 48 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/CMUC_2007__48_2_a6/ LA - en ID - CMUC_2007__48_2_a6 ER -
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/