Algebraic & Geometric Topology

Universal nowhere dense subsets of locally compact manifolds

Taras Banakh and Dušan Repovš

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In each manifold M modeled on a finite- or infinite-dimensional cube [0,1]n, nω, we construct a closed nowhere dense subset SM (called a spongy set) which is a universal nowhere dense set in M in the sense that for each nowhere dense subset AM there is a homeomorphism h:MM 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 AA and B of these decompositions are ambiently homeomorphic.

Article information

Algebr. Geom. Topol., Volume 13, Number 6 (2013), 3687-3731.

Received: 8 February 2012
Accepted: 21 May 2013
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N20: Topology of infinite-dimensional manifolds [See also 58Bxx] 57N40: Neighborhoods of submanifolds
Secondary: 57N45: Flatness and tameness 57N60: Cellularity

Universal nowhere dense subset Sierpiński carpet Menger cube Hilbert cube manifold $n$–manifold tame ball tame decomposition


Banakh, Taras; Repovš, Dušan. Universal nowhere dense subsets of locally compact manifolds. Algebr. Geom. Topol. 13 (2013), no. 6, 3687--3731. doi:10.2140/agt.2013.13.3687.

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