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Math Help - open set

  1. #1
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    open set

    Are the following are open sets in . In each case say why?

    1.[0,)

    2. -


    3.

    For 1. I know its not an open set because the complement is open so [0,) is closed.


    But need help for 2 and 3.

    Thanks
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by 1234567 View Post
    Are the following are open sets in . In each case say why?

    1.[0,)

    2. -


    3.

    For 1. I know its not an open set because the complement is open so [0,) is closed.


    But need help for 2 and 3.

    Thanks
    How are you defining open here? the natural way? open subsets of the reals are open intervals? if so,

    yes, 1. is closed, because its compliment \displaystyle (-\infty, 0) is open.

    For 2, notice that you have the set:

    \displaystyle \dots (-2, -1) \cup (-1,0) \cup (0,1) \cup (1,2) \cup \dots

    For 3, \displaystyle \mathbb Q is a set of points on the real line... but be careful, \mathbb Q is dense in \mathbb R, so you're going to want to come from the perspective of boundary points, etc
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  3. #3
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    Yes open subsets of the reals are open intervals.

    Thanks for the help.

    For 3. I know the complement is the set of all irrational numbers which is not open so do that mean \displaystyle \mathbb Q is also not open.
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  4. #4
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    Quote Originally Posted by 1234567 View Post
    Yes open subsets of the reals are open intervals.
    Actually any open set is is the union of open intervals.
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