| Title:
|
Covering $\Sigma^0_\xi$-generated ideals by $\Pi^0_\xi$ sets (English) |
| Author:
|
Mátrai, Tamás |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
48 |
| Issue:
|
2 |
| Year:
|
2007 |
| Pages:
|
245-268 |
| . |
| Category:
|
math |
| . |
| Summary:
|
\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. (English) |
| Keyword:
|
Borel $\sigma$-ideal |
| Keyword:
|
Hurewicz test |
| MSC:
|
03E15 |
| MSC:
|
54H05 |
| idZBL:
|
Zbl 1199.54189 |
| idMR:
|
MR2338093 |
| . |
| Date available:
|
2009-05-05T17:02:41Z |
| Last updated:
|
2012-05-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119655 |
| . |
| Reference:
|
[1] Jech T.: Set Theory.Springer Monographs in Mathematics, Springer, Berlin, 2003. Zbl 1007.03002, MR 1940513 |
| Reference:
|
[2] Kechris A.S.: Classical Descriptive Set Theory.Graduate Texts in Mathematics 156, Springer, New York, 1995. Zbl 0819.04002, MR 1321597 |
| Reference:
|
[3] Louveau A., Saint Raymond J.: Borel classes and closed games: Wadge-type and Hurewicz-type results.Trans. Amer. Math. Soc. 304 2 (1987), 431-467. Zbl 0655.04001, MR 0911079 |
| Reference:
|
[4] Mátrai T.: Hurewicz tests: separating and reducing analytic sets on the conscious way.PhD Thesis, Central European University, 2005. |
| Reference:
|
[5] Mátrai T.: ${\boldsymbol \Pi}^{0}_{2}$-generated ideals are unwitnessable.submitted for publication. |
| Reference:
|
[6] Miller A.: Problems.http://www.math.wisc.edu/ miller/res/problem.pdf. Zbl 1160.90358 |
| Reference:
|
[7] Solecki S.: Covering analytic sets by families of closed sets.J. Symbolic Logic 59 3 (1994), 1022-1031. Zbl 0808.03031, MR 1295987 |
| Reference:
|
[8] Solecki S.: Decomposing Borel sets and functions and the structure of Baire class $1$ functions.J. Amer. Math. Soc. 11 3 (1998), 521-550. Zbl 0899.03034, MR 1606843 |
| . |
