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$\sigma$-porosity; descriptive set theory; $\sigma$-ideal; trigonometric series; sets of uniqueness
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.
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