Descriptive properties of \(\sigma\)-porous sets (Q2501024)

The author constructs two \(\sigma\)-porous sets with strange covering properties. For a set \(E \subseteq \mathbb R \), let \(P(E)\) stand for the set of porosity points of \(E\), i.e.\ \[ x \in P(E) \iff \exists \varepsilon > 0~ \forall R > 0 ~\exists 0 < r < R ~\exists y \in (x-r, x+r) \colon (y-\varepsilon r , y+ \varepsilon r ) \cap E = \emptyset. \] The author constructs a closed set \(H \subseteq \mathbb R \) such that \(P(H)\) can be covered by no \(F_{\sigma \delta}\) \(\sigma\)-porous set; and also a non-\(\sigma\)-porous perfect nowhere dense set \(L \subseteq \mathbb R \) such that \(P(L)\) is \(G_{\delta}\). The first result is a counterpart of \textit{J. Foran} and \textit{P. D. Humke} [Real. Anal. Exch. 6, 114--119 (1981; Zbl 0467.28001)] stating that each \(\sigma\)-porous set can be covered by a \(G_{\delta \sigma}\) \(\sigma\)-porous set. The second result is a counterpart of a construction of \textit{J. Tkadlec} [ibid. 9, 473--482 (1984; Zbl 0582.28001)] for a porous set of the form \(P(H)\), where \(H\) is a suitable nowhere dense perfect non-\(\sigma\)-porous set, without \(\sigma\)-porous \(G_{\delta}\) envelope. The constructions are heavily based on the results of \textit{M. Zelený} and \textit{J. Pelant} [Commentat. Math. Univ. Carol. 45, No. 1, 37--72 (2004; Zbl 1101.28001)] and \textit{L. Zajíček} [Colloq. Math. 77, No. 2, 293-304 (1998; Zbl 0909.28001)].