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Keywords:
universal nowhere dense subset; Sierpiński carpet; Menger cube; Hilbert cube manifold; $n$-manifold; tame ball; tame decomposition
universal nowhere dense subset; Sierpiński carpet; Menger cube; Hilbert cube manifold; $n$-manifold; tame ball; tame decomposition
Summary:
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.
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