I am trying to show that $\mathbb{R}^2-\{0,0\}$ is not an extendible manifold. We can consider the universal cover $M$ with covering map $\pi$, and suppose for contradiction that $M\subset M'$ is an extension. Then we pick a point $p\in \partial M$, and consider a neighborhood $W'\subset M'$, a (convex) neighborhood of $p$.
What I'm reading claims that if $W'-\{p'\}\subset M$, we obtain a contradiction since then $\pi(W'-\{p'\})=U$ is a neighborhood of $(0,0)$ in $\mathbb{R}^2$; we then lift up a circle in $M$, a contradiction.
What I'm mainly confused about is how to prove that $\pi(W'-\{p'\})$ is indeed a neighborhood of $(0,0)$. It makes sense intuitively, since the boundary of $\partial M$ should map to the point $(0,0)$, but I'm not exactly sure how to show this rigorously. Why does this neighborhood have to include $(0,0)$ and a region surrounding it? It seems like the result should depend on the $p$ we pick.
