$B=B(0,1)\subset\mathbb{R}^N$ and $\Omega=B\setminus\{0\}$.
(i) Assume $N=1$ and prove $H^1_0(\Omega)\neq H^1_0(B)$.
(ii) Take $N\ge 2$. Does $H^1_0(\Omega)=H^1_0(B)$?
I don't even know where to start with. I think one can probably use the fact that $H_0^1(\Omega)=\overline{C_0^\infty(\Omega)}^{H^1(\Omega)}$ to construct sequences in order to give a counterexample or prove the statement. But the detail is really beyond me. Any insight would be helpful. Thank you very much!