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Math Help - Countable union of closed sets/countable interesection of open sets

  1. #1
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    Countable union of closed sets/countable interesection of open sets

    A set A is called an F_\sigma set if it can be written as the countable union of closed sets. A set B is called a G_\delta set if it can be written as the countable intersection of open sets.
    (a) Show that a closed interval [a, b] is a G_\delta set.
    (b) Show that the half open interval (a, b] is both a G_\delta and an F_\sigma set.
    (c) Show that Q is an F_\sigma set, and the set of irrationals I forms a G_\delta set.

    I have no idea where to start on this problem. Any help would be appreciated. Thank you for your time.
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  2. #2
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    For (a) consider the family of open sets

    \displaystyle A_n=\left( a-\frac{1}{n},b+\frac{1}{n}\right)

    What happens when you take the intersection of

    \displaystyle \cap_{n=1}^{\infty}A_n?

    This should get you started on (a) and (b)

    Also remember that in \mathbb{R} points are closed and \mathbb{Q} is a countable set.

    Good luck.
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  3. #3
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    I have decided to show you the irrational part of part #3.
    Let \mathbb{I}=\mathbb{R}\setminus\mathbb{Q} the irrational numbers.

    The let \{x\}^c=\left( { - \infty ,x} \right) \cup \left( {x,\infty } \right).

    Try to convince yourself that  \mathbb{I}= \bigcap\limits_{x \in \mathbb{Q}} {\left\{ x \right\}^c } , a G_{\delta} set.
    Last edited by Plato; October 8th 2010 at 01:31 PM.
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  4. #4
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    Thanks all! You guys rock!
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