The structure of the \(\sigma \)-ideal of \(\sigma \)-porous sets. (English) Zbl 1101.28001

Zelený, Miroslav; Pelant, Jan

Commentat. Math. Univ. Carol. 45, No. 1, 37-72 (2004).

Summary: We show a general method for constructing 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 \(\mathcal K(E)\), then it is shown that each analytic subset of \(\mathcal K(E)\) 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.

MSC:

28A05	Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
26E99	Miscellaneous topics in real functions
42A63	Uniqueness of trigonometric expansions, uniqueness of Fourier expansions, Riemann theory, localization
54H05	Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)

Keywords:

\(\sigma \)-porosity; descriptive set theory; trigonometric series; sets of uniqueness