Pairwise disjoint Moebius bands in space

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 $R^3$ 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 $R^N$, $N\geqslant 3$. Second, we show that in case of $R^3$ the theorem holds for arbitrarily topologically embedded (not necessarily tame) Moebius bands.