No Embedding From A Torus to A Sphere

Suppose that there exists an embedding $f : T^2 \mapsto S^2$.

Each point $x \in T^2$ has an open neighborhood $U$ homeomorphic to the open unit disc in $S^2$, and it follows that $f(U) \subset S^2$ is homeomorphic to the open unit disc. By the Invariance of Domain theorem, $f(U)$ is an open subset of $S^2$. This shows that $f(T^2)$ is an open subset of $S^2$.

But $T^2$ is compact, so $f(T^2)$ is compact, so it is also a closed subset of $S^2$.

By connectivity of $S^2$, it follows that $f(T^2)=S^2$. So $f$ is a homeomorphism, contradicting that $S^2$ is simply connected and $T^2$ is not.

Here are given several reasons why the torus cannot be embedded into $\mathbb R^2$; two of them use the invariance of domain theorem.

Now, if the torus could be embedded into $S^2$, then this embedding cannot be onto $S^2$, as otherwise this would be a homeomorphism. Thus, as $S^2$ minus a point is homeomorphic to $\mathbb R^2$, we would get an embedding of the torus into $\mathbb R^2$.