# Chart(s) of space-time as a smooth manifold

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?

• Spacetime is covered by a single metric-preserving chart if and only if you are studying special relativity. – WillO Feb 17 '19 at 5:25
• Thanks, I understand now, indeed I was carrying stereotypes from special relatively – pig2000 Feb 17 '19 at 5:49
• As an example I found atlas of de sitter space: arxiv.org/abs/1211.2363 – pig2000 Feb 17 '19 at 5:55
• @WillO: Spacetime is covered by a single metric-preserving chart if and only if you are studying special relativity. The part about "metric-preserving" isn't relevant to the OP. – user4552 Feb 17 '19 at 15:22
• @BenCrowell: I agree that it is not relevant to the question that the OP actually asked, but I am at least 70% confident that it is relevant to the underlying confusion. – WillO Feb 17 '19 at 16:38