Question: let $X=\mathbb{R}$ with topology generated by open intervals of the form $(a,b)$and sets of the form $\mathbb{Q}∩(a,b)$ then $X$ is not normal space.
My attempt: I know if for each pair of disjoint closed sets $A$ and $B$, if there exists open sets $U$ and $V$ such that, $A⊆U$ , $B⊆V$ and $U∩V≠∅$ then space will be normal and complements of sets in topology are closed sets in space. But, however I am unable to find such disjoint pair of closed sets and unable to prove if such pair exists then, how to show they are not separate. I am beginner in topology, please help me..:-(
Try separating the closed set $\mathbb{P} = \mathbb{R}\setminus \mathbb{Q}$ from the closed set $\{0\}$.