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I've found really good answers for the question from link below.

Any open subset of $\Bbb R$ is a at most countable union of disjoint open intervals. [Collecting Proofs]

But I have a question about how $\mathbb{R}$, which is open subset of $\mathbb{R}$, can be expressed as union of countable collection of disjoint open intervals. In my intuition, it seems not possible.... Thanks for your help.

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Open intervals in $\Bbb R$ are allowed to extend to $\infty$, so $(-\infty,\infty)$ is one open interval that is all of $\Bbb R$.

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  • $\begingroup$ Oh...so infinite end point is possible. Thank you!! $\endgroup$
    – twnzre
    Commented Dec 25, 2019 at 14:47
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    $\begingroup$ In one sense it is not an end point, because the interval is open. The use of $\infty$ means there is no end point. $\endgroup$ Commented Dec 25, 2019 at 14:49
  • $\begingroup$ Oh that's right Thanks for correction $\endgroup$
    – twnzre
    Commented Dec 25, 2019 at 14:53

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