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Keywords:
$\sigma $-porous sets; $\sigma $-ideal; coanalytic sets; Hausdorff metric
$\sigma $-porous sets; $\sigma $-ideal; coanalytic sets; Hausdorff metric
Summary:
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.
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