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$\sigma $-porous sets; $\sigma $-ideal; coanalytic sets; Hausdorff metric
Let $\bold P$ be a porosity-like relation on a separable locally compact metric space $E$. We show that the $\sigma$-ideal of compact $\sigma$-$\bold P$-porous subsets of $E$ (under some general conditions on $\bold P$ and $E$) forms a $\boldsymbol \Pi_{\bold 1}^{\bold 1}$-complete set in the hyperspace of all compact subsets of $E$, in particular it is coanalytic and non-Borel. Our general results are applicable to most interesting types of porosity. It is shown in the cases of the $\sigma$-ideals of $\sigma$-porous sets, $\sigma$-$\langle g \rangle$-porous sets, $\sigma$-strongly porous sets, $\sigma$-symmetrically porous sets and $\sigma$-strongly symmetrically porous sets. We prove a similar result also for $\sigma$-very porous sets assuming that each singleton of $E$ is very porous set.
[DP] Debs G., Preiss D.: private communication.
[DSR] Debs G., Saint-Raymond J.: Ensemblesboréliens d'unicité au sens large. Ann. Inst. Fourier (Grenoble) 37 3 217-239 (1987). MR 0916281
[EH] Evans M.J., Humke P.D.: Contrasting symmetric porosity and porosity. J. Appl. Anal. 4 1 19-41 (1998). MR 1648939 | Zbl 0922.26003
[F] Foran J.: Continuous functions need not have $\sigma$-porous graphs. Real Anal. Exchange 11 194-203 (1985-86). MR 0828490 | Zbl 0607.26005
[K$_1$] Kechris A.S.: Classical Descriptive Set Theory. Springer-Verlag, New York, 1995. MR 1321597 | Zbl 0819.04002
[K$_2$] Kechris A.S.: On the concept of $\boldsymbol \Pi_{\bold 1}^{\bold 1}$-completeness. Proc. Amer. Math. Soc. 125 6 1811-1814 (1997). MR 1372034
[KL] Kechris A.S., Louveau A.: Descriptive Set Theory and the Structure of Sets of Uniqueness. London Math. Soc. Lecture Notes Series 128, Cambridge University Press, Cambridge, 1989. MR 0953784 | Zbl 0677.42009
[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
[M] Michael E.: Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71 (1951), 152-182. MR 0042109 | Zbl 0043.37902
[Za$_1$] Zajíček L.: Sets of $\sigma$-porosity and $\sigma$-porosity $(q)$. Časopis Pěst. Mat. 101 (1976), 350-359. MR 0457731
[Za$_2$] Zajíček L.: Porosity and $\sigma$-porosity. Real Anal. Exchange 13 (1987-88), 314-350. MR 0943561
[Za$_3$] Zajíček L.: Products of non-$\sigma$-porous sets and Foran systems. Atti Sem. Mat. Fis. Univ. Modena 44 (1996), 497-505. MR 1428780
[Za$_4$] Zajíček L.: Smallness of sets of nondifferentiability of convex functions in non-separable Banach spaces Czechoslovak Math. J. 41 (116) (1991), 288-296. MR 1105445
[Zam] Zamfirescu T.: Nearly all convex bodies are smooth and strictly convex. Monatsh. Math. (1987), 57-62. MR 0875352 | Zbl 0607.52002
[Ze] Zelený M.: unpublished manuscript.
[ZP] Zelený M., Pelant J.: The structure of the $\sigma$-ideal of $\sigma$-porous sets. submitted, available electronically under MR 2076859
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