| Title:
|
A remark on a theorem of Solecki (English) |
| Author:
|
Holický, P. |
| Author:
|
Zajíček, L. |
| Author:
|
Zelený, M. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
46 |
| Issue:
|
1 |
| Year:
|
2005 |
| Pages:
|
43-54 |
| . |
| Category:
|
math |
| . |
| Summary:
|
S. Solecki proved that if $\Cal F$ is a system of closed subsets of a complete separable metric space $X$, then each Suslin set $S\subset X$ which cannot be covered by countably many members of $\Cal F$ contains a $\boldsymbol G_{\delta}$ set which cannot be covered by countably many members of $\Cal F$. We show that the assumption of separability of $X$ cannot be removed from this theorem. On the other hand it can be removed under an extra assumption that the $\sigma $-ideal generated by $\Cal F$ is locally determined. Using Solecki's arguments, our result can be used to reprove a Hurewicz type theorem due to Michalewski and Pol, and a nonseparable version of Feng's theorem due to Chaber and Pol. (English) |
| Keyword:
|
Solecki's theorem |
| Keyword:
|
Suslin set |
| Keyword:
|
$\sigma$-ideal |
| MSC:
|
03E15 |
| MSC:
|
28A05 |
| MSC:
|
54H05 |
| idZBL:
|
Zbl 1121.03058 |
| idMR:
|
MR2175858 |
| . |
| Date available:
|
2009-05-05T16:49:29Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119507 |
| . |
| Reference:
|
[B] Baire R.: Sur la représentation des fonctions discontinues.Acta Math. 30 (1905), 1-48. MR 1555022 |
| Reference:
|
[CP] Chaber J., Pol R.: Remarks on closed relations and a theorem of Hurewicz.Topology Proc. 22 (1997), 81-94. Zbl 0943.54018, MR 1657906 |
| Reference:
|
[E] Engelking R.: General Topology.Heldermann, Berlin, 1989. Zbl 0684.54001, MR 1039321 |
| Reference:
|
[F] Feng Q.: Homogeneity for open partitions of pairs of reals.Trans. Amer. Math. Soc. 339 (1993), 659-684. Zbl 0795.03065, MR 1113695 |
| Reference:
|
[KLW] Kechris A.S., Louveau A., Woodin W.H.: The structure of $\sigma $-ideals of compact sets.Trans. Amer. Math. Soc. 301 (1987), 263-288. Zbl 0633.03043, MR 0879573 |
| Reference:
|
[K] Kunen K.: Set Theory. An Introduction to Independence Proofs.Springer, New York, 1980. Zbl 0534.03026, MR 0597342 |
| Reference:
|
[Ku] Kuratowski K.: Topology.vol. I, Academic Press, New York, 1966. Zbl 0849.01044, MR 0217751 |
| Reference:
|
[L] Lusin N.N.: Collected Works, Part $2$.Moscow, 1958 (in Russian). |
| Reference:
|
[MP] Michalewski H., Pol R.: On a Hurewicz-type theorem and a selection theorem of Michael.Bull. Polish Acad. Sci. Math. 43 (1995), 273-275. Zbl 0841.54029, MR 1414783 |
| Reference:
|
[P] Petruska Gy.: On Borel sets with small covers.Real Anal. Exchange 18 (1992-93), 330-338. MR 1228398 |
| Reference:
|
[S] Solecki S.: Covering analytic sets by families of closed sets.J. Symbolic Logic 59 (1994), 1022-1031. Zbl 0808.03031, MR 1295987 |
| Reference:
|
[Z] Zajíček L.: On $\sigma$-porous sets in abstract spaces (a partial survey).Abstr. Appl. Anal., to appear. MR 2201041 |
| Reference:
|
[ZZ] Zajíček L., Zelený M.: Inscribing closed non-$\sigma$-lower porous sets into Suslin non-$\sigma$-lower porous sets.Abstr. Appl. Anal., to appear. MR 2197116 |
| . |
