#### Volume 13, issue 6 (2013)

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Universal nowhere dense subsets of locally compact manifolds

### Taras Banakh and Dušan Repovš

Algebraic & Geometric Topology 13 (2013) 3687–3731
##### Abstract

In each manifold $M$ modeled on a finite- or infinite-dimensional cube ${\left[0,1\right]}^{n}$, $n\le \omega$, 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 $h:\phantom{\rule{0.3em}{0ex}}M\to M$ such that $h\left(A\right)\subset S$. The key tool in the construction of spongy sets is a theorem on the topological equivalence of certain decompositions of manifolds. A special case of this theorem says that two vanishing cellular strongly shrinkable decompositions $\mathsc{A},\mathsc{ℬ}$ of a Hilbert cube manifold $M$ are topologically equivalent if any two nonsingleton elements $A\in \mathsc{A}$ and $B\in \mathsc{ℬ}$ of these decompositions are ambiently homeomorphic.

##### Keywords
Universal nowhere dense subset, Sierpiński carpet, Menger cube, Hilbert cube manifold, $n$–manifold, tame ball, tame decomposition
##### Mathematical Subject Classification 2010
Primary: 57N20, 57N40
Secondary: 57N45, 57N60