Let $n,k\in\mathbb N$ with $1\le k\le n$. A $k$-dimensional ($C^1$-) submanifold of $\mathbb R^n$ is a non-empty set $M\subseteq\mathbb R^n$ with the property that for all $a\in M$ there exists an open set $\Omega\in\mathbb R^n$, an open set $T\subseteq\mathbb R^k$ and there is an immersion $\varphi\in C^1(T,\mathbb R^n)$, such that $a\in\Omega$, $\varphi(T)=M\cap\Omega$, and $\varphi$ is a homeomorphism from $T$ to $M\cap\Omega$. Then $\varphi$ is called a chart.
Given the $S^2$ sphere, does a chart $\varphi$ of $S^2$ exist with $S^2=\varphi(T)$ ? If not, why?
No, for otherwise $\varphi$ would define a homeomorphism $T\to S^2$ where $T$ is open in $\Bbb R^k$. This, however, is impossible since $S^2$ is compact and $T$ is not compact.