$A^\circ \cup B^\circ \subset (A \cup B )^\circ$ Counterexample for = instead of $\subset$

Question

If $A$ and $B$ are sets of real numbers, then $(A \cup B)^{\circ} \supseteq A^ {\circ}\cup B^{\circ}$

But the same relation with a = isn't always true. Can someone find an example where the = doesn't hold, I can't seem to find one.

Thanks in advance.

Answer 1

$A = \mathbb{Q}, B = \mathbb{R} \setminus \mathbb{Q}$. Then $\operatorname{Int}(A) = \operatorname{Int(B)} = \emptyset$, while $\operatorname{Int}(A \cup B) = \operatorname{Int}(\mathbb{R}) = \mathbb{R}$. This is the most extreme example of what can go wrong...

Answer 2

You can take $A=(-\infty,0]$ and $B=[0,\infty)$, as subspaces of $\mathbb R$ with the usual topology.