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Universal nowhere dense subsets of locally compact manifolds

Taras Banakh and Dušan Repovš

Algebraic & Geometric Topology 13 (2013) 3687–3731

In each manifold M modeled on a finite- or infinite-dimensional cube [0,1]n, n ω, we construct a closed nowhere dense subset S M (called a spongy set) which is a universal nowhere dense set in M in the sense that for each nowhere dense subset A M there is a homeomorphism h: M M such that h(A) 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 A, of a Hilbert cube manifold M are topologically equivalent if any two nonsingleton elements A A and B of these decompositions are ambiently homeomorphic.

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
Received: 8 February 2012
Accepted: 21 May 2013
Published: 16 October 2013
Taras Banakh
Department of Geometry and Topology
Ivan Franko National University of Lviv
1 Universytetska str
Lviv, 79000
Instytut Matematyki
Jan Kochanowski University in Kielce
15 Swietokrzyska str
25-406 Kielce
Dušan Repovš
Faculty of Education and Faculty of Mathematics and Physics
University of Ljubljana
Kardeljeva ploscad 16
1000 Ljubljana