I learned that the following two definitions of nowhere dense set are equivalent.
(1) $E$ is nowhere dense in $X$ if the closure of $E$ in $X$ has an empty interior.
(2) For any open set $V\subset X$, the set $V \cap E$ is NOT dense in $V$ (in the subspace topology).
I am able to show that (1) implies (2), but I am not able to show (2) implies (1).
My attempt: Assuming (2), I assumed to the contrary that there is an non-empty open set $J$ such that $J \subset closure(A)$. Then, I tried to show $J\cap E$ is dense in $J$, but I am stuck.
How should I proceed?