So we all know that space-time in general relativity is modeled as a smooth (pseudoRiemannian) manifold.
Each point (event) on space-time is labeled with a unique coordinate $(t,x,y,z)$ in a specific reference frame.
Am I correct to think the map
$spacetime \rightarrow \mathbb{R}^4$
$event \rightarrow (t,x,y,z)$
as a chart on the entirety of the manifold?
Hence did we just assumed that space-time is homeomorphic to $\mathbb{R}^4$ so that a single chart can cover the entirety of the manifold?
What if space-time is homeomorphic to for example $S^4$ so that one single chart isn't enough?