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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?

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  • $\begingroup$ Take any set that is neither open nor closed, and its complement. $\endgroup$ – celtschk Mar 14 '18 at 7:01
  • $\begingroup$ $\Bbb R$ 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. $\endgroup$ – DanielWainfleet Mar 14 '18 at 7:06
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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.

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

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