Universal nowhere dense and meager sets in Menger manifolds

Abstract

In each Menger manifold $M$ we construct: (i) a closed nowhere dense subset $M_0$ which is homeomorphic to $M$ and is universal nowhere dense in the sense that for each nowhere dense set $A\subset M$ there is a homeomorphism $h$ of $M$ such that $h(A)\subset M_0$; (ii) a meager $F_\sigma$-set $\Sigma_0\subset M$ which is universal meager in the sense that for each meager subset $B\subset M$ there is a homeomorphism $h$ of $M$ such that $h(B)\subset \Sigma_0$. Also we prove that any two universal meager $F_\sigma$-sets in $M$ are ambiently homeomorphic.