$\mathbb{R}$ is union of countable collection of disjoint open intervals

Question

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.

score 5 · Accepted Answer · answered Dec 25 '19 at 14:42

5

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

answered Dec 25 '19 at 14:42

Ross Millikan

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Oh...so infinite end point is possible. Thank you!! – twnzre Dec 25 '19 at 14:47
1

In one sense it is not an end point, because the interval is open. The use of $\infty$ means there is no end point. – Ross Millikan Dec 25 '19 at 14:49
Oh that's right Thanks for correction – twnzre Dec 25 '19 at 14:53

$\mathbb{R}$ is union of countable collection of disjoint open intervals

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