It is a well-known fact that any open set $O$ of real numbers may be written as the pairwise disjoint union of countably many open intervals $I_n$.

However, I am wondering: when is it possible to say that $O$ is a finite union of pairwise disjoint open intervals?

If $O$ is an interval, then this is trivial.

Are there more obvious/standard cases when we can say this for an open set $O$?


iff $O$ is a union of finitely many open intervals, but I suppose you know that already.

Maybe you are looking for something like $O^c\cap\overline{O}$ is finite?


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