Definition of a nowhere dense set

I'm currently studying metric spaces through Gamelin and Greene's Introduction to Topology. While studying about completeness I got stuck with this concept of nowhere dense subset. The book defines a subset Y of X is nowhere dense if cl(Y) has no interior points i.e. int(cl(Y)) is empty. It then says that "evidently Y is nowhere dense if and only if X\cl(Y) is a dense open subset of X". This is the part I don't get. How can I prove the iff relationship between this statement and the original definition of a nowhere dense set?

$$X \setminus \operatorname{int} A = \operatorname{cl} (X\setminus A),$$
Thus $\overline{Y}$ has empty interior if and only if $X \setminus \overline{Y}$ is dense.
• Note that $x \in X \setminus \operatorname{int}A$ iff it is not the case that there is an open set containing $x$ that stays inside $A$. So every open set that contains $x$ also contains points not in $A$. And this just says that $x \in \operatorname{cl}(X \setminus A)$. Hence the equality. – Henno Brandsma Mar 14 '14 at 6:25