Universal nowhere dense and meager sets in Menger manifolds

Abstract

In each Menger manifold M we construct:•a closed nowhere dense subset M0 which is homeomorphic to M and is universal nowhere dense in the sense that for each nowhere dense set A⊂M there is a homeomorphism h of M such that h(A)⊂M0;•a meager Fσ-set Σ0⊂M which is universal meager in the sense that for each meager subset B⊂M there is a homeomorphism h of M such that h(B)⊂Σ0. Also we prove that any two universal meager Fσ-sets in M are ambiently homeomorphic.