# Article

Full entry | PDF   (0.2 MB)
Keywords:
Solecki's theorem; Suslin set; $\sigma$-ideal
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.
References:
[B] Baire R.: Sur la représentation des fonctions discontinues. Acta Math. 30 (1905), 1-48. MR 1555022
[CP] Chaber J., Pol R.: Remarks on closed relations and a theorem of Hurewicz. Topology Proc. 22 (1997), 81-94. MR 1657906 | Zbl 0943.54018
[E] Engelking R.: General Topology. Heldermann, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[F] Feng Q.: Homogeneity for open partitions of pairs of reals. Trans. Amer. Math. Soc. 339 (1993), 659-684. MR 1113695 | Zbl 0795.03065
[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. MR 0879573 | Zbl 0633.03043
[K] Kunen K.: Set Theory. An Introduction to Independence Proofs. Springer, New York, 1980. MR 0597342 | Zbl 0534.03026
[Ku] Kuratowski K.: Topology. vol. I, Academic Press, New York, 1966. MR 0217751 | Zbl 0849.01044
[L] Lusin N.N.: Collected Works, Part $2$. Moscow, 1958 (in Russian).
[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. MR 1414783 | Zbl 0841.54029
[P] Petruska Gy.: On Borel sets with small covers. Real Anal. Exchange 18 (1992-93), 330-338. MR 1228398
[S] Solecki S.: Covering analytic sets by families of closed sets. J. Symbolic Logic 59 (1994), 1022-1031. MR 1295987 | Zbl 0808.03031
[Z] Zajíček L.: On $\sigma$-porous sets in abstract spaces (a partial survey). Abstr. Appl. Anal., to appear. MR 2201041
[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

Partner of