Covering -generated ideals by sets
Commentationes Mathematicae Universitatis Carolinae (2007)
- Volume: 48, Issue: 2, page 245-268
- ISSN: 0010-2628
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topMátrai, Tamás. "Covering $\Sigma ^0_\xi $-generated ideals by $\Pi ^0_\xi $ sets." Commentationes Mathematicae Universitatis Carolinae 48.2 (2007): 245-268. <http://eudml.org/doc/250230>.
@article{Mátrai2007,
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 $\{\Pi \}^\{0\}_\{\xi \}$ and not $\{\Sigma \}^\{0\}_\{\xi \}$ subset $P$ of a Polish space $X$ there is a $\sigma $-ideal $\mathcal \{I\}\subseteq 2^\{X\}$ such that $P\notin \mathcal \{I\}$ but for every $\{\Sigma \}^\{0\}_\{\xi \}$ set $B\subseteq P$ there is a $\{\Pi \}^\{0\}_\{\xi \}$ set $B^\{\prime \}\subseteq P$ satisfying $B\subseteq B^\{\prime \}\in \mathcal \{I\}$. We also discuss several other results and problems related to ideal generation and Hurewicz test pairs.},
author = {Mátrai, Tamás},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Borel $\sigma $-ideal; Hurewicz test; -ideal; Baire category; Hurewicz test},
language = {eng},
number = {2},
pages = {245-268},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Covering $\Sigma ^0_\xi $-generated ideals by $\Pi ^0_\xi $ sets},
url = {http://eudml.org/doc/250230},
volume = {48},
year = {2007},
}
TY - JOUR
AU - Mátrai, Tamás
TI - Covering $\Sigma ^0_\xi $-generated ideals by $\Pi ^0_\xi $ sets
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 2
SP - 245
EP - 268
AB - 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 ${\Pi }^{0}_{\xi }$ and not ${\Sigma }^{0}_{\xi }$ subset $P$ of a Polish space $X$ there is a $\sigma $-ideal $\mathcal {I}\subseteq 2^{X}$ such that $P\notin \mathcal {I}$ but for every ${\Sigma }^{0}_{\xi }$ set $B\subseteq P$ there is a ${\Pi }^{0}_{\xi }$ set $B^{\prime }\subseteq P$ satisfying $B\subseteq B^{\prime }\in \mathcal {I}$. We also discuss several other results and problems related to ideal generation and Hurewicz test pairs.
LA - eng
KW - Borel $\sigma $-ideal; Hurewicz test; -ideal; Baire category; Hurewicz test
UR - http://eudml.org/doc/250230
ER -
References
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- Miller A., Problems, http://www.math.wisc.edu/ miller/res/problem.pdf. Zbl1160.90358
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