Universal nowhere dense and meager sets in Menger manifolds

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 $F_σ$-set $Σ_0\subset M$ which is universal meager in the sense that for each meager subset $B\subset M$ there is a homeomorphism $h$ of $M$ such that $h(B)\subset Σ_0$. Also we prove that any two universal meager $F_σ$-sets in $M$ are ambiently homeomorphic.

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