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?



Does every compact manifold exhibit an almost global chart
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. ) 


