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I'm currently tackling the following question which has two parts:

"Can you write the set $[0,1]$ as a countable intersection of open intervals. Either find suitable real numbers $a_1, a_2,....,b_1, b_2...$ so that

$[0,1] = \bigcap_{n \in \mathbb{n}} (a_n, b_n)$

Or argue why this is impossible"

The second section essentially asks the same question but as a union instead of intersection.

I'm very new to this topic and just have no idea where to start, any proofs / help is much appreciated.

Thanks

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  • $\begingroup$ First part: this is possible, think of an example. Second part: this is impossible, look at the endpoints. $\endgroup$ – Wojowu Oct 23 '17 at 18:33
  • $\begingroup$ If it is the intersection, you must have $[0,1]$ in every open interval. What does that say about $a_n,b_n$? The union of any collection of open sets is open. $\endgroup$ – copper.hat Oct 23 '17 at 18:37
  • $\begingroup$ Any union (countable or not) of open sets is open. Since $[0,1]$ is not open, you won't be able to express it as a union of open sets (including open intervals). $\endgroup$ – Bungo Oct 23 '17 at 18:38
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For the first part take $a_n=-\frac{1}{n}$, $b_n=1+\frac{1}{n}$ and prove this choice works.

For the second part: prove that your union of intervals must be an open interval (so it cannot be closed).

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  • 2
    $\begingroup$ Why not just ${1 \over n}$ instead of ${1 \over n+1}$? You contributing to the global symbol shortage. $\endgroup$ – copper.hat Oct 23 '17 at 18:41
  • $\begingroup$ Good point. I didn't see it started at 1. $\endgroup$ – Keen Oct 23 '17 at 18:56
  • $\begingroup$ Just continuing the global fight against symbol wastage :-). $\endgroup$ – copper.hat Oct 23 '17 at 19:02

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