Compact differentiable sub manifold with at least two points

Question

Let $M$ be a differentiable submanifold of $\mathbb R^n$ which contains at least two points. How can I show that if $M$ is compact in $\mathbb R^n$ there exists no atlas for $M$ which only consists of one chart?

Answer 1

Suppose that the dimension of $M$ is $p$, the existence of such a chart is equivalent to saying that there exists a connected open subset $U$ of $R^p$ and a homeomorphism $f:U\rightarrow M$, since $M$ is compact, $U$ has to be compact. But a non empty open subset of $R^p,p>0$ is not compact.