Lud\v ek Zaj\'{\i }\v cek, Miroslav Zelen\'y
On the complexity of some $\sigma $-ideals of $\sigma $-P-porous sets

Comment.Math.Univ.Carolinae 44,3 (2003) 531-554.

Abstract:Let $\bold P$ be a porosity-like relation on a separable locally compact metric space $E$. We show that the $\sigma $-ideal of compact $\sigma $-$\bold P$-porous subsets of $E$ (under some general conditions on $\bold P$ and $E$) forms a $\boldsymbol \Pi _{\bold 1}^{\bold 1}$-complete set in the hyperspace of all compact subsets of $E$, in particular it is coanalytic and non-Borel. Our general results are applicable to most interesting types of porosity. It is shown in the cases of the $\sigma $-ideals of $\sigma $-porous sets, $\sigma $-$\langle g \rangle $-porous sets, $\sigma $-strongly porous sets, $\sigma $-symmetrically porous sets and $\sigma $-strongly symmetrically porous sets. We prove a similar result also for $\sigma $-very porous sets assuming that each singleton of $E$ is very porous set.

Keywords: $\sigma $-porous sets, $\sigma $-ideal, coanalytic sets, Hausdorff metric
AMS Subject Classification: 54H05, 28A05