TITLE: Does every compact manifold exhibit an almost global chart
QUESTION [15 upvotes]: Let $M$ be a compact connected manifold.
Is there a chart $\Psi:U \to \mathbb{R}^n$ such that the closure of $U$ is $M$?
This is true for $S^n, T^n, K$, all compact surfaces, etc.
If it is not true in general, what is the obstruction?

REPLY [6 votes]: The exponential map for any  Riemannian metric on your compact manifold $M$, based at any point $p$ of $M$,  maps the tangent space $T_p M$, an ${\mathbb R}^n$, onto $M$ and  is a diffeo inside the cut locus.   Back on the tangent space, this `inside' of  the  cut locus is a  star shaped domain relative to the origin, so defines a domain $V$ in ${\mathbb R}^n$ which is mapped diffeomorphically onto an open set $U$ whose closure is $M$. 
(The closure of the domain $V$ is homeomorphic to the closed ball in the tangent space, so this same argument shows that every compact manifold is the quotient of the n-ball by some identification of points on its boundary, the n-sphere. )