# How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?

I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded into the Euclidean space of the same dimension. So I wonder if all manifolds can be embedded into a same dimensional Euclidean space when we cut out a measure zero set? Can anyone prove it or disprove it by giving me some counterexamples?

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+1, interesting question! Another way of phrasing it is, how big can we make a chart? I tried to make a title that was easier to parse, but feel free to change it to something else if you want. –  Zev Chonoles Mar 6 at 1:35
@Zev: In your phrasing, the question is a duplicate of both math.stackexchange.com/questions/11769/… and also of math.stackexchange.com/questions/18083/…. I'm not sure what the policy is on closing because something is provable equivalent to a duplicate ;-). –  Jason DeVito Mar 6 at 1:46
One way to get at this question is to put a Riemannian metric on the manifold and then to consider how big the cut locus of a point can be. I believe it is known that the cut-locus of a point has measure zero, although I forget the proof. If this is the case, you could always remove a measure zero set and then use the exponential map at a point to get the required embedding. –  treble Mar 6 at 1:46
What is cut locus? –  lee Mar 6 at 13:16
Another idea would be to take a Morse function with a unique minimum. Then, take the union of all the ascending manifolds from higher index critical points. This is of codimension 1 and thus of measure 0. I'm pretty sure the complement is the ascending manifold of the minimum and thus diffeomorphic to a ball. Oh, wait, that's the link that Jason gave. Sorry. –  Sam Lisi Mar 20 at 13:39