Proof verification: interior of a nowhere dense set is empty

Question

A set $S$ in a topological space $X$ is called nowhere dense if its exterior is dense in $X$.

Now suppose that $A$ is a nowhere dense set. We have $Int(A)= X-Cl(A^c)=X-X=\emptyset$. Is it correct?

Answer 1

Yes, it is correct.

You can also prove it saying that since $\mathring A$ is an open set, $\mathring{A}^\complement$ is closed and, since it contains $A^\complement$, $\mathring{A}^\complement\supset\overline{A^\complement}=X$. Therefore, $\mathring{A}^\complement=X$ and therefore $\mathring A=\emptyset$. But it is basically the same idea.