Is $\mathbb R-\mathbb Q$ open or closed?

Question

The set $\Bbb R-\Bbb Q$ is open, closed or neither? And how would one prove it?

I tried to prove that $\Bbb Q$ is open, so $\Bbb R-\Bbb Q$ would be closed, but I am not sure that $\Bbb Q$ is open.

Answer 1

If you take $x \in \Bbb Q$, and any $\epsilon > 0$, then $(x-\epsilon, x+ \epsilon)$ contains points of both $\Bbb Q$ and $\Bbb R - \Bbb Q$. Meaning that $(x-\epsilon,x+\epsilon) \not\in \Bbb Q$. So $\Bbb Q$ can't be open, and $\Bbb R - \Bbb Q$ can't be closed. So both of them are neither.

Answer 2

These sets are neither open nor closed. Moreover,

$$ cl (\Bbb Q) = \Bbb R,\qquad cl (\Bbb R\setminus\Bbb Q)=\Bbb R,$$ where $cl(A)$ denotes the closure of $A$.