Interior of a set $A$ is dense in $X$ iff $A$ is dense in $X$ I would like to check if the following statement is true: 

Interior of a set $A$ is dense in $X$ iff $A$ is dense in $X$

It seems to me that they are equivalent, but I couldn't prove or find a counterexample for it.
Thanks!

Similarly, another example is that the irrationals ${\mathbb R}\setminus{\mathbb Q}$ are dense in $\mathbb R$ but have empty interior.
