# Does the 'closure of the interior' equal the 'interior of the closure'?

My answer is no because, $\mathbb{Q}^o = \emptyset$ and so $\overline{(\mathbb{Q}^o)} = \emptyset$ but $\overline{\mathbb{Q}} = \mathbb{R}$ and so $\big(\overline{\mathbb{Q}}\,\big)^o = \mathbb{R}$.

Is my example correct?

-
This is the first example that comes to mind. – Norbert May 13 '12 at 11:09
Yes, your example is correct. – Rudy the Reindeer May 13 '12 at 11:12
It doesn't need to be so ill-behaved though. For example, take $(0,1)$ in $\mathbb{R}$. The closure of the interior is $[0,1]$, but the interior of the closure is $(0,1)$. – Chris Eagle May 13 '12 at 11:19
The title is somewhat misleading. Note that the closure of any set is closed, while the interior of any set is open. The only two sets (in $\mathbb R$) which are both closed and open are the empty set and $\mathbb R$. – Asaf Karagila May 13 '12 at 11:32
Attention The examples are corrects if you have usual topology – diofanto May 13 '12 at 11:35