This question refers to How to prove closure of $\mathbb{Q}$ is $\mathbb{R}$
I want to prove that the closure of $\mathbb{Q}$ is $\mathbb{R}$. I am trying to understand the accepted answer, but when it comes to "This shows that the complement of $\mathbb{Q}$ has empty interior, so the closure of $\mathbb{Q}$ is all of $\mathbb{R}$.", I get stuck.
What does it mean intuitively that a set has empty interior?
What I know is that a point is interior to a set when it is the center of some open ball inside that set. In that sense, for a set to have empty interior would mean that it has no interior points. Furthermore, the set of all its adherent points - a point $m$ is adherent to the set of rationals when there is a sequence of rational points $x_k$ such that $limx_k = m.$Being so, the complement of rationals would have no interior points.
How does it follow from this that the closure of $\mathbb{Q}$ is $\mathbb{R}$?
Thanks in advance!