I am learning Sobolev spaces. There seem to be a difference while approximating a function in $W^{k,P}(\Omega)$ by smooth function $C^\infty (\Omega)$ and $C^\infty ( \overline \Omega)$, where we use $\overline \Omega$ to refer to the closure of $\Omega$.
How does the closure make a difference , waiting for explanation.