Manifolds, charts and coordinates Let's consider the manifold $S^1$
It is well known that we need two charts to cover this manifold.
Nonetheless, we can cover the full space using a single coordinate $\theta$ which is just the angle from the center.
Now, is this a general feature? I mean, is it always possible to have in every manifold a single coordinate set that cover points that are in different charts, just as in $S^1$?

Perhaps you're "really" asking whether or not every $n$-manifold $M$ has $\mathbf{R}^{n}$ as a covering space, i.e., whether there's a single (smooth) map $\pi:\mathbf{R}^{n} \to M$ whose restrictions to suitably small subsets define coordinate charts on $M$.
If so, the answer is "no"; even among spheres, only $S^{1}$ has this property.
