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Is this a closed set?

  1. Sep 17, 2009 #1
    The set [itex]\{1,2,3,4,5\ldots\}[/itex]...is it closed as a subset of [itex]\mathbb{R}[/itex]? I'm thinking "yes," but I'm unsure of myself for some reason. (And yes, this is just the set of positive integers.
     
  2. jcsd
  3. Sep 17, 2009 #2

    Office_Shredder

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    The definition of closed is the complement of an open set. So is the set

    [tex]( - \infty,1) \cup (1,2) \cup (2,3) \cup (3,4)...[/tex] open?
     
  4. Sep 17, 2009 #3
    Can you show that its complement is open in [itex]\mathbb{R}[/itex]?
     
  5. Sep 17, 2009 #4
    A closed set contains its bondary.
    The boundary are al the points where you can draw an arbitrarily small circle around it and there are always points inside and outside the set. So yes your set is closed and it is its own boundary.
     
    Last edited: Sep 17, 2009
  6. Sep 17, 2009 #5
    Yes. Any union of open sets is open. Thanks! :)
     
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