Embedding of $2$-torus minus one point into $\mathbb{R}^2$ as an open set?

Question

Let $M^2$ be the $2$-torus minus one point. Is there an embedding of $M^2$ into $\mathbb{R}^2$ as an open set?

Answer 1

By stereographic projection we can let this embedding be to $S^2$ instead. Let $B$ be a small closed disk centered at the deleted point in $T^2 -\{p\}$, and let $\gamma = \partial B$. By the Jordan curve theorem, the image of $\gamma$ under such an embedding separates $S^2$ into two disks, and therefore the image of $T^2-int(B)$ under the embedding is a closed and open subset of a closed disk and hence is a closed disk. But $\pi_1(T^2-int(B)) \cong F_2$ the free group on two generators so such an embedding never exists.