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Thread: Intersection of finitely many open sets

  1. #1
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    Intersection of finitely many open sets

    Hello all,

    I am asked to argue for or against the following statement:

    The interval $\displaystyle \left[0,1\right]$ is an intersection of countably many open sets of $\displaystyle \mathbb{R}$

    I have the following considerations:

    1) The intersection of countably many opens sets produces an open set. Since $\displaystyle \left[0,1\right]$ is closed then it can not be an intersection of countably many opens sets.

    2) I consider the interval $\displaystyle \left]-\frac{1}{n},1+\frac{1}{n}\right[$. Then $\displaystyle \bigcap_{n=1}^{\infty} \left]-\frac{1}{n},1+\frac{1}{n}\right[ = \left[0,1\right] $

    Based on point 1) and 2) I would conclude that $\displaystyle \left[0,1\right]$ is NOT an intersection of countably many open sets but IS an intersection of infinitely many open sets.

    Is the above considerations and conclusion correct?

    Thanks.
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  2. #2
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    Re: Intersection of finitely many open sets

    Quote Originally Posted by detalosi View Post
    I am asked to argue for or against the following statement:
    The interval $\displaystyle \left[0,1\right]$ is an intersection of countably many open sets of $\displaystyle \mathbb{R}$
    I have the following considerations:
    1) The intersection of countably many opens sets produces an open set. Since $\displaystyle \left[0,1\right]$ is closed then it can not be an intersection of countably many opens sets.
    2) I consider the interval $\displaystyle \left]-\frac{1}{n},1+\frac{1}{n}\right[$. Then $\displaystyle \bigcap_{n=1}^{\infty} \left]-\frac{1}{n},1+\frac{1}{n}\right[ = \left[0,1\right] $
    Based on point 1) and 2) I would conclude that $\displaystyle \left[0,1\right]$ is NOT an intersection of countably many open sets but IS an intersection of infinitely many open sets. Is the above considerations and conclusion correct? NO ABSOLUTELY NOT!
    You have confused union with intersection.
    The union of open sets is open. The intersection of finitely many open sets is open.
    Thanks from topsquark and detalosi
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  3. #3
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    Re: Intersection of finitely many open sets

    $$\underset{n=1}{\overset{\infty }{\bigcap }}\left(-\frac{1}{n},1+\frac{1}{n}\right)=[0,1]$$

    that's the intersection of countably many open sets (intervals) $= [0,1]$

    it is also the intersection of infinitely many open sets

    $[0,1]$ cannot be written as the finite intersection of open sets because that would be an open set

    but $[0,1]$ is not an open set
    Thanks from detalosi
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