DSE 2020 Question 36: Real analysis

Let $\mathcal{B}$ be the collection of sets $E\subseteq\mathbb{R}$ such that for every $x\in E$ there are real numbers $a<b$ with $x\in(a,b)\subseteq E$. Let $\mathcal{C}$ be the collection of sets $E\subseteq\mathbb{R}$ such that for every $x\in E$ there are rational numbers $a<b$ with $x\in(a,b)\subseteq E$. Then