Let $L$ be the anti-diagonal of $\Bbb R_\ell^2$, $D=\Bbb Q^2$, $A=L\cap D$ and $B=L\setminus D$ and suppose $V$ is an open set of $X$ containing $B$.
For each $n\in\Bbb N_+$ we define $$K_n=\{x\in[0,1]\setminus\Bbb Q:\exists n [x,x+1/n)\times [-x,-x+1/n)\subseteq V\}.$$
For part (a) it's see to see that $\bigcup K_n=[0,1]\setminus\Bbb Q$
Part of this question has been answered here: Proof that Sorgenfrey plane is not normal using points x × (-x).
However, I'm stuck on a different part of the problem, namely (b). This part refers to Exercise 5 of Section 27. It states, that if $X$ is compact, Hausdorff and $\{A_n\}$ is a family of closed sets with empty interior, then $\bigcup A_n$ has empty interior.
The idea is to use the contrpositive of this statement to show that there is a $\overline K_n$ that contains an open interval. From part (a) we see that $\overline{\bigcup K_n}=[0,1]$, but we are considering $\bigcup\overline K_n$. In general, $\bigcup\overline K_n\subseteq\overline{\bigcup K_n}$. How do we know that $\bigcup\overline K_n$ contains an open interval?
