# Packing moebius bands

I know that in the smooth category the following is true. There are at most countable many embedded moebius bands in euclidean 3-space. Is this also true in topological category?

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What exactly are you counting? Isotopy classes of single embeddings, or perhaps the number of disjoint embeddings? – Ryan Budney Aug 6 '11 at 17:04
Disjont embeddings. – Michal Aug 6 '11 at 17:09
How do you know this in the smooth category? – Igor Rivin Aug 6 '11 at 20:48
A few years ago I came across a paper by Grushin and Palamodov from 1962 called "On the maximal number of mutually disjoint, pairwise homeomorphic figures which can be packed in 3-space" (Uspekhi Mat. Nauk, 1962, Volume 17, Issue 3(105), Pages 163–168) where the case of Möbius strips is considered. Unfortunately the paper is in Russian and I know of no translations so I wasn't able to read it...and I am also not sure if this answers the OP's question or it's just the case he affirms to know already...It would be great if someone could tell me some details about what that paper says! – godelian Aug 7 '11 at 1:49

There are at most countably many disjoint embeddings of homeomorphic images of a non-orientable hypersurface in $\mathbb R^k$. This is theorem 2 in "An uncountable family of disjoint spatial continua in Euclidean space" by V.K. Ionin and Yu.G. Nikonorov.