Dense subset of $L^2(\Omega)$

Question

Consider the function space $A =L^2(\Omega)$, where $\Omega\subseteq \mathbb{R}^n$ is open. We have seen that the space of test functions, $B=C_0^{\infty}(\Omega)$ is dense in $A$. Now my question is,

what about the space $S=C^{\infty}_0(\Omega \setminus \{a\})$ for some element $a$ in $\Omega$, e.g., $a=0$? How can I determine whether $S$ is dense in $A$? All the proofs I have seen showing $\overline{B}=A$ are constructive and use carefully constructed functions from $B$ as well as convolutions, and I am not sure how to translate this into my case, if possible at all.

Any help would be appreciated.

Answer 1

You are getting unnecsesarily confused! $C_0^{\infty} (\Omega) \subset S$ which makes $S$ obviously dense.