Universal nowhere dense subsets of locally compact manifolds

Banakh, Taras¹ ; Repovš, Dušan²

¹Department of Geometry and Topology, Ivan Franko National University of Lviv, 1 Universytetska str, Lviv, 79000, Ukraine, and, Instytut Matematyki, Jan Kochanowski University in Kielce, 15 Swietokrzyska str, 25-406 Kielce, Poland
²Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, Kardeljeva ploscad 16, 1000 Ljubljana, Slovenia

Algebraic and Geometric Topology, Tome 13 (2013) no. 6, pp. 3687-3731

Résumé

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,ℬ of a Hilbert cube manifold M are topologically equivalent if any two nonsingleton elements A ∈A and B ∈ℬ of these decompositions are ambiently homeomorphic.

Détail

DOI : 10.2140/agt.2013.13.3687

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

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