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Theorem: Every non-empty open set S in $\mathbb{R^1}$ is the union of a countable collection of disjoint component intervals of S.

I think it is quite easy to prove that the component intervals is disjoint but i am not sure how to do for countable union.

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    [possible duplicate](http://math.stackexchange.com/questions/19066/show-that-a-subset-a-of-r-is-open-iff-it-is-countable-union-of-open-subsets).2012-07-04

3 Answers 3