
TL;DR
This paper extends a classical result by proving that uncountably many disjoint Moebius bands cannot be embedded in space, generalizing to tame subsets in higher dimensions and to arbitrary embeddings in three-dimensional space.
Contribution
It generalizes Grushin and Palamodov's 1962 theorem to tame subsets in R^N and arbitrary embeddings of Moebius bands in R^3.
Findings
Uncountably many disjoint Moebius bands cannot be embedded in R^3.
The result extends to tame subsets in R^N for N ≥ 3.
The theorem holds for arbitrary topological embeddings in R^3.
Abstract
V.V.Grushin and V.P.Palamodov proved in 1962 that it is impossible to place in uncountably many pairwise disjoint polyhedra each homeomorphic to the Moebius band. We generalize this result in two directions. First, we give a generalization of this result to tame subsets in , . Second, we show that in case of the theorem holds for arbitrarily topologically embedded (not necessarily tame) Moebius bands.
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