Some properties of (Φ)-uniformly symmetrically porous sets

Abstract

We prove that each perfect linear set contains a perfect set which is (\(\Phi\))-uniformly symmetrically porous (Theorem 1). In the hyperspace of all nonempty compact sets in \([0,1]\) (endowed with the Hausdorff distance), the (\(\Phi\))-uniformly symmetrically porous nonempty compact sets form a \(G_\delta\) residual subspace (Theorem 2). We infer that the\linebreak (\(\Phi\))-uniformly symmetrically porous perfect sets form a \(G_\delta\) residual set in the space of all perfect sets in \([0,1]\) (Theorem 3).