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Confusion with Disconnected sets

  1. May 8, 2013 #1
    Hello,

    I am having some difficulties understanding why a subset under the usual metric topology of the reals is connected.

    How can a set X = (0,1] u (1,2) be connected?

    The definition I am using is:

    A is disconnected if there exists two open sets G and V and the following three properties hold:

    (1) A intersect G ≠ ∅
    A intersect V ≠ ∅

    (2) A is a proper subset of the union of G and V.

    (3) the intersection of G and V is the empty set.

    Thanks
     
    Last edited: May 8, 2013
  2. jcsd
  3. May 8, 2013 #2

    micromass

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    And you also want ##G## and ##V## to be open. No?
     
  4. May 8, 2013 #3
    Yea, that would be more correct.
     
  5. May 13, 2013 #4

    HallsofIvy

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    Okay, so can you find two such open sets for [itex](0, 1]\cup (1, 2)[/itex]? (0, 1] and (1, 2) will not do because (0, 1] is not open. (And, did you notice that [itex](0, 1]\cup (1, 2)= (0, 2)[/itex]?)
     
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