Universal nowhere dense subsets of locally compact manifolds

Abstract

In each manifold $M$ modeled on a finite or infinite dimensional cube $[0,1{]}^n$ we construct a closed nowhere dense subset $S\subset M$ (called a spongy set) which is a universal nowhere dense set in $M$ in the sense that for each nowhere dense subset $A\subset M$ there is a homeomorphism $h:M\to M$ such that $h(A)\subset S$. The key tool in the construction of spongy sets is a theorem on topological equivalence of certain decompositions of manifolds. A special case of this theorem says that two vanishing cellular strongly shrinkable decompositions $\mathcal{A},\mathcal{B}$ of a Hilbert cube manifold $M$ are topologically equivalent if any two non-singleton elements $A\in \mathcal{A}$ and $B\in \mathcal{B}$ of these decompositions are ambiently homeomorphic.

24 pages

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• Universal nowhere dense and meager sets in Menger manifolds