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Homework Help: Connected Space Proof

  1. Feb 18, 2012 #1
    1. The problem statement, all variables and given/known data
    Show that X is connected if and only if the only subsets of X that are both open and closed are the empty set and X.

    Proof: https://files.nyu.edu/eo1/public/Book-PDF/Appendix.pdf [Broken]
    Page 14.

    I'm confused by this proof. First, if S is not in {null set, X} then how can S be a subset of X?
    Secondly, how can X = S U (X\S)?
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Feb 18, 2012 #2

    Office_Shredder

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    Suppose that X is the set of real numbers. Then we could let S be the set [0,1]. {null,X} contains only two elements: the null set, and the set of real numbers, neither of which are [0,1]
     
  4. Feb 18, 2012 #3
    OK thanks. I didn't consider X to be a single element, then any nonempty subset of X would not be in {null, X}
     
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