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Set S=Q$\cap$(0,1) I wanted to show that this can not be expressed as intersection of countable collection of open sets.
I know Any closed set can be shown as intersection of countable collection of open set .
I had hint show this by Cantor intersection property which says that infinite intersection of nested sequence of nonempty compact set is nonempty . I could not able to link this hint with problem till now .Any help will be appreciated



Without using Baire Category Theorem Is it possible to solve this problem? As this problem occur in exercise where there in no mention of that theorem in text.

Curious student
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  • Do you know what the Baire category theorem is ? –  May 03 '18 at 16:33
  • "I wanted to show that this can not be expressed as intersection of countable open sets." This is not true. – Logic_Problem_42 May 03 '18 at 16:34
  • @harmonicuser I don't know about Baire Category Theorem – Curious student May 03 '18 at 16:42
  • @Logic_Problem_42 Sorry Sir But This Problem in Tom Apostol Mathematical Analysis Page 66 problem 3.16 – Curious student May 03 '18 at 16:44
  • Then I was wrong. Look here: https://math.stackexchange.com/questions/206559/a-question-about-finding-a-countable-collection-of-open-sets-where-intersection – Logic_Problem_42 May 03 '18 at 16:45
  • But it's interesting, why my proof is false. I would enumerate the rational numbers from $(0,1)$: $a_1,a_2,....$ and then I would define open sets as $A_n=\cup_{k=1}^{\infty} (a_k-\frac{1}{n2^k},a_k+\frac{1}{n2^k})$. – Logic_Problem_42 May 03 '18 at 16:48
  • Suppose on the contrary that $S=\bigcap\nolimits_{i\in I}{{{O}_{i}}}$.

    (Where $I$ is a countable index set and ${{O}_{i}}$ is open for each $i$ ),

    Choose a rational number to be in this countable intersection, you need an open interval containing it in every $i$ . But every such interval also contains irrational number.

     –  May 03 '18 at 16:50
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    @CheerfulParsnip, this doesn't seem to be a duplicate. The other question specifically asks why a specific example is not a counter example. Not how to prove it. Moreover, this question specifically asks about how not to use the Baire theorem. – user56834 May 03 '18 at 18:02
  • @Programmer2134 I would say it is a duplicate, since one of the answers to the linked question does not use BCT, as desired. – Mike Earnest May 03 '18 at 18:35

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