TITLE: Is an immersion an embedding almost everywhere?
QUESTION [10 upvotes]: Let $M, N$ be smooth manifolds, and let $F \colon M \to N$ be a smooth immersion. I know that $F$ is a local embedding, but is it also an embedding almost everywhere?
In other words, does there exists a set of measure zero $X \subset M$ such that $F$ restricted to $M \setminus X$ is a smooth embedding?

REPLY [14 votes]: In case it's of interest, even an injective immersion can fail everywhere to be an embedding. An irrational winding on a torus is an example: Fix an irrational number $\alpha$. The path $\gamma(t) = (t, \alpha t)$ in the real plane descends to an injective regular path on the square torus $(\mathbf{R}/\mathbf{Z})^{2}$ whose image is dense.