I want to prove that the two dimensional disk $D^2$ is not a topological manifold without boundary, i.e. there is a point $x\in D^2$ such that $x$ has no neighbourhood $U$ with $U$ homeomorphic to $\mathbb{R}^2$.
I choose $x=1\in\partial D^2$. Without loss of generality, let $U$ be a contractible open neighbourhood and suppose $U$ is homeomorphic to $\mathbb{R}^2$ with $f:U\to\mathbb{R}^2$ a homeomorphism. Is it true that $U\setminus\{1\}$ is contractible? And what is the reason it is so?
Why not pick $U$ to be a small open disk intersected with $D^2$? E.g., $U = B_\epsilon(x) \cap D^2$. Then $U\setminus \{x\}$ is contractible because it is star-shaped.