Pairwise disjoint Moebius bands in space

Abstract

V. V. Grushin and V. P. Palamodov proved in 1962 that it is impossible to place in [Formula: see text] 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 [Formula: see text], [Formula: see text]. Second, we show that in case of [Formula: see text] the theorem holds for arbitrarily topologically embedded (not necessarily tame) Moebius bands.