Do 2 sets exist that are neither open nor closed, but their union is both open and closed.

Question

I'm under the impression that only 2 clopen sets exist ${\rm I\!R}$ and $\emptyset$?

So does there exist 2 sets neither open or closed that form one of these clopen sets?

Answer 1

Yes, the set of rational numbers and the set of irrational numbers are neither open nor closed but the union is the set of real numbers which is both open and closed.

Answer 2

Sure. Consider: $$ A = (-\infty, 3) \cup [4, 7] $$ and $$ B = [3, 4) \cup (7, \infty) $$