Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Confusion with Disconnected sets

  1. May 8, 2013 #1

    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.

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


    User Avatar
    Science Advisor

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

Similar Discussions: Confusion with Disconnected sets
  1. Open set (Replies: 2)

  2. Connected set (Replies: 16)

  3. Union of sets (Replies: 3)

  4. Open set (Replies: 19)