Does there always exist a continuous/smooth map from $\mathbb{R}^n$ onto a manifold $M^n$? I want to know if we can get away with only having one set of coordinates for a manifold if we allow multiple coordinates to map to the same point.

Another construction goes through Riemannian metrics. Assume that $M$ is connected. If you put a complete riemannian metric on your manifold, the exponential map $\exp_p:T_pM\rightarrow M$ will be surjective. The idea is that any two points are connected by a geodesic. You can view the world standing in one point!
