1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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}
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Connected Space Proof
  1. Connected Spaces (Replies: 3)

Loading...