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?
Best Answer
Because they are both dense (proved in real analysis) and are disjoint (by definition).
Whenever $A$ and $B$ are dense disjoint subsets of a topological space $X$, we have $\overline A=X=\overline B$ by the definition of being dense. Since $B\subset A^c$ and $A\subset B^c$, it follows that $\overline{A^c}=X=\overline{B^c}$. The closure of complement is the complement of interior; hence, both interiors are empty. The conclusion about boundary follows by removing interior from the closure: $X\setminus \varnothing=X$.