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I am struggling to understand the proof from the book of Adams and Hedberg, "Function spaces and potential theory". It seems to me that there is a serious flaw, and moreover, the statement is ...

Recall that a set $E \subset \mathbb{R}^n$ is called (1,2) stable if it satisfies the following: $$ W_0^{1,2}(E) = W_0^{1,2}(E^0), $$ where $E^0$ is the interior of $E$ and $W_0^{1,2}(E)$ is defined ...