Abstract:In each manifold $M$ modeled on a finite or infinite dimensional cube $[0,1]^n$, $n\leq \omega $, we construct a meager $F_\sigma$-subset $X\subset M$ which is universal meager in the sense that for each meager subset $A\subset M$ there is a homeomorphism $h:M\to M$ such that $h(A)\subset X$. We also prove that any two universal meager $F_\sigma$-sets in $M$ are ambiently homeomorphic.
Keywords: universal nowhere dense subset, Sierpi\'nski carpet, Menger cube, Hilbert cube manifold, $n$-manifold, tame ball, tame decomposition
AMS Subject Classification: 57N20 57N45 54F65