I came across this statement in one of the introductions to differential geometry that some of the manifolds cannot be expressed with a single global coordinate system and one of the examples is the surface of a sphere.
A global coordinate system is one where we have one-to-one mapping from all points on manifold (S) to $\mathbb R^n$. I can map each point on a sphere to $\mathbb R^3$, hence I should have a global coordinate system?
Am I missing something very obvious?
A chart for a manifold $M$ ($\dim M = n$) is a pair $(U,\varphi)$ where $U\subseteq M$ is open and $\varphi\colon U \to \varphi[U]\subseteq \Bbb R^n$ is a homeomorphism onto the open set $\varphi[U] \subseteq \Bbb R^n$. If $M$ is compact, the existence of a global chart $(M,\varphi)$ means that the image $\varphi[M] \subseteq \Bbb R^n$ is open, non-empty, and compact (by continuity of $\varphi$). This is impossible.