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If a set A is both open and closed then it is R(set of real numbers)

  1. Sep 10, 2009 #1
    if a set A is both open and closed then it is R(set of real numbers) how we may show it in a proper way
     
  2. jcsd
  3. Sep 10, 2009 #2
    Re: theorem

    First, are you talking about an open-and-closed subset of R? If you are, then the empty set is also open-and-closed in R, so you have to specify that A is nonempty.

    My approach for this proof would be to consider this.

    An open set contains none of its boundary points.
    A closed set contains all its boundary points.

    The only way for this to be possible is for A to have NO boundary points at all. Show how {} and R are the only two sets that have this property.
     
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