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.

10 pages

References in corpus (1)

• Universal nowhere dense subsets of locally compact manifolds