Show there exists a set $E \subset \mathbb{R}^2$ such that the cross sections $[E]_a$ and $[E]^a$ are open subsets for every $a \in \mathbb{R}$ but $E \notin \mathcal{B}_2$.
This is a question from Axler's book on pg 144, exercise 2 and $\mathcal{B}_2$ is the $\sigma$-algebra of Borel subsets of $\mathbb{R}^n$.
I know that in general, if a set $E \in \mathbb{R}^n$ is open, the cross sections aren't generally open that if the cross sections are open, then it isn't necessarily true that the set itself is open.
The condition that $E$ can't be in the Borel $\sigma$-algebra is really bothering me. I also thought maybe trying to use the Cantor function generalized to $\mathbb{R}^n$, but didn't see immediately how to keep cross sections as open subsets.
Let $A$ be a non-Borel subset of $\mathbb R$. Then $$ E = \mathbb R^2 \setminus \{(x,x): x \in A\} $$ is your set. It is not Borel. But all the cross-sections are either $\mathbb R$ or $\mathbb R$ minus a single point.