Consider $\mathbb Q$, the set of rational numbers, and its complement $\mathbb R\setminus \mathbb Q$, the set of irrational numbers.
I noticed that their interiors, closures and boundaries are the same, that is:
- Interior: $\varnothing$
- Closure: $\Bbb R$
- Boundary: $\Bbb R$
Why does this happen? Is this a part of some general pattern?