Frontier properties of a subset of $\mathbb{R}$

Question

I am supposed to prove or disprove the following claims:

1. If $A \subset \mathbb{R}$ arbitrary, then $(Fr(Fr(A))^{\mathbb{o}} = \emptyset$
2. If $A, B \subset \mathbb{R}$ arbitrary, then $Fr(A \times B) = (Fr(A) \times B) \cup (A \times Fr(B))$

It seems the latter claim is correct, but I cannot prove it. It's not hard to visualize. On the other hand, I have no idea about the first claim.

Answer 1

• the boundary of a closed set or an open set has empty interior.

• the boundary of any set is closed.

Together this implies that $\operatorname{int}(\partial \partial A) = \emptyset$ for any $A$, the first boundary operation makes it closed and then the boundary of a closed set has empty interior.

Fact 2 is refuted by $A = B =\mathbb{Q}$, as $(\mathbb{R} \times \mathbb{Q}) \cup (\mathbb{Q} \times \mathbb{R}) \neq \mathbb{R} \times \mathbb{R}$. IIRC, we need a term $\partial A \times \partial B$ in the union as well.