Geodesic
The structure of the $\sigma$-ideal of $\sigma$-porous sets
Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 1, pp. 37-72 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We show a general method of construction of non-$\sigma$-porous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each non-$\sigma$-porous Suslin subset of a topologically complete metric space contains a non-$\sigma$-porous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a non-$\sigma$-porous element. Namely, if we denote the space of all compact subsets of a compact metric space $E$ with the Vietoris topology by $\Cal K(E)$, then it is shown that each analytic subset of $\Cal K(E)$ containing all countable compact subsets of $E$ contains necessarily an element, which is a non-$\sigma$-porous subset of $E$. We show several applications of this result to problems from real and harmonic analysis (e.g. the existence of a closed non-$\sigma$-porous set of uniqueness for trigonometric series). Finally we investigate also descriptive properties of the $\sigma$-ideal of compact $\sigma$-porous sets.
We show a general method of construction of non-$\sigma$-porous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each non-$\sigma$-porous Suslin subset of a topologically complete metric space contains a non-$\sigma$-porous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a non-$\sigma$-porous element. Namely, if we denote the space of all compact subsets of a compact metric space $E$ with the Vietoris topology by $\Cal K(E)$, then it is shown that each analytic subset of $\Cal K(E)$ containing all countable compact subsets of $E$ contains necessarily an element, which is a non-$\sigma$-porous subset of $E$. We show several applications of this result to problems from real and harmonic analysis (e.g. the existence of a closed non-$\sigma$-porous set of uniqueness for trigonometric series). Finally we investigate also descriptive properties of the $\sigma$-ideal of compact $\sigma$-porous sets.
Classification : 26E99, 28A05, 42A63, 54H05
Keywords: $\sigma$-porosity; descriptive set theory; $\sigma$-ideal; trigonometric series; sets of uniqueness 
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     title = {The structure of the $\sigma$-ideal of $\sigma$-porous sets},
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Zelený, Miroslav; Pelant, Jan. The structure of the $\sigma$-ideal of $\sigma$-porous sets. Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 1, pp. 37-72. http://geodesic.mathdoc.fr/item/CMUC_2004_45_1_a3/