[math-fun] bicoloring the plane (or any manifold) so all curves bicolored

Somebody a few months ago wanted a way to 2-color the plane so that no continuous curve could be monochromatic. My solution, which works in any-dimensional space: Pick two dense subsets A and B of the reals R, with A subset B subset R. [Example B=terminating decimal expansion, A=and the last digit is 1.] If X has either all coordinates in complement(B) or at least one coordinate in A, then color red. Else blue. This actually works on any Riemannian manifold of any dimensionality by using the Nash embedding theorem http://en.wikipedia.org/wiki/Nash_embedding_theorem