Open Set Examples in Real Numbers

Question

I cannot understand why $\mathbb {R\setminus Q}$ is a $G_{\delta}$ set in $(-\infty,0]$

I can’t imagine any open set in real numbers except intervals. Which open sets’ intersection is Irrational numbers? $(\le 0)$

Thanks

Answer 1

Hint: Take the open sets $(-\infty,0]\setminus\{r\} $.

Answer 2

With regards to your first question, rationals are clearly $F_\sigma$ (singletons are closed, and $\mathbb{Q}$ is countable), and it is straightforward to show from DeMorgan's laws that the complement of an $F_\sigma$ set is $G_\delta$.

EDIT: To elaborate, let $A$ be an arbitrary $F_\sigma$ set, and write it as $\bigcup_{n=1}^\infty A_n$, where each $A_n$ is closed. Take the complement and apply DeMorgan's laws.