Definition of a nowhere dense set

Question

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?

Answer 1

It follows from the relation

$$X \setminus \operatorname{int} A = \operatorname{cl} (X\setminus A),$$

"the complement of the interior is the closure of the complement". (Prove that relation, if you're not yet familiar with it.)

Thus $\overline{Y}$ has empty interior if and only if $X \setminus \overline{Y}$ is dense.