Let $A$ be a subset of a topological space $X$.
(1) $A$ is nowhere dense if the interior of its closure is empty;
(2) $A$ has dense interior if the closure of its interior is $X$;
(3) $A$ has empty interior if its interior is empty;
(4) $A$ is dense if its closure is $X$.
Now, to have dense interior (= to be dense and open) is stronger than to be dense.
I have a doubt: Is to be nowhere dense stronger than to have empty interior?