Sorry, if this is a stupid question. But then I'm not good in real analysis either. I came across to a definition ("Length of Sets"), whose starting was "Let $O$ be an open set in $\mathbb{R}$. Then $O$ can be written as a countable union of mutually disjoint open intervals..."
My doubts :
What do they mean by Open Sets? If they are talking in one-space, can't they use " Let $O$ be an open INTERVAL in $\mathbb{R}$. Then $O$ can be written as a countable union of mutually disjoint open intervals.."
By "$O$ can be written as a countable union of mutually disjoint open intervals..", do they mean, for example, $O=(1,2)\cup(2,3)\cup(3,4)$.. If this is, then how can $O$ be an open set(interval) as $2$ doesn't belong to $O$.