I am studying Carroll's notes on GR. He defines charts as maps from an open subset in manifold [itex]M[/itex] to open subset in [itex]R^n[/itex]. He then writes "We therefore see the necessity of charts and atlases: many manifolds cannot be covered with a single coordinate system. (Although some can, even ones with nontrivial topology. Can you think of a single good coordinate system that covers the cylinder, S1 × R?)" So when we have defined the chart as a map whose domain is an open subset in the manifold then how can a single chart cover the entire manifold? Or am I missing something?