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

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?

• Take any set that is neither open nor closed, and its complement. – celtschk Mar 14 '18 at 7:01
• $\Bbb R$ and $\emptyset$ are the only open-and-closed sets in the usual topology on $\Bbb R.$ This is because it is a connected space. In general if $A$ is a subset of a space $S$, and $A$ is neither open nor closed, let $B=S\setminus A.$ Then $B$ is not open or closed. But $A\cup B=S$, which is open and closed. – DanielWainfleet Mar 14 '18 at 7:06

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