  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.
