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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 :

  1. 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.."

  2. 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 $O$.

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  • $\begingroup$ By definition, an open set is a union of open intervals. These may be disjoint, for example $(0,1) \cup (3,5)$ is an open set. $\endgroup$ – Crostul Aug 23 '16 at 12:41
  • $\begingroup$ @Crostul, thanks for commenting. I read about Open sets on mathworld.wolfram.com/OpenSet.html as Wikipedia has a long article on it. There they have defined it as "In one-space, the open set is an open interval . In two-space, the open set is a disk . In three-space, the open set is a ball." But they didn't mention it as an union of open intervals. $\endgroup$ – user117741 Aug 23 '16 at 12:49
  • $\begingroup$ That is an incorrect definition of open set. On the real line an open set is a union of open intervals. I think the way they worded it was confusing. $\endgroup$ – D_S Aug 23 '16 at 12:50
  • $\begingroup$ @D_S, Oh, right then, I'll check Wikipedia. $\endgroup$ – user117741 Aug 23 '16 at 13:14
  • $\begingroup$ @Fel thanks, any book which you would recommend me to start? $\endgroup$ – user117741 Aug 23 '16 at 13:16
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An open subset of $\Bbb R$ is a set $O$ with the property such that for each $x\in O$ there exists $\epsilon>0$ such that the open interval $(x-\epsilon,x+\epsilon)$ is contained (as subset) in $O$. The open set $O$ itself need not be an interval.

Note hat $(1,2)\cup (2,3)\cup (3,4)$ is an open set in the above sense, but is not an open interval. (It is of course, by the very form it is presented as, the union of finitely - hence countably - many mutually disjoint open intervals)

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