Miroslav Zelen\'{y}, Jan Pelant
The structure of the $\sigma $-ideal of $\sigma $-porous sets

Comment.Math.Univ.Carolinae 45,1 (2004) 37-72.

Abstract: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.

Keywords: $\sigma $-porosity, descriptive set theory, $\sigma $-ideal, trigonometric series, sets of uniqueness
AMS Subject Classification: 28A05, 26E99, 42A63, 54H05