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Topology question

  1. May 12, 2008 #1
    I'm struggling with something that I suspect is very basic. How do I should that the closure of a connected set is connected? I think I need to somehow show that it is not disconnected, but that's where I'm stuck.

  2. jcsd
  3. May 13, 2008 #2
    Call your space [tex]M[/tex]. You want to show that if [tex]\mathrm{cl}(M) \subset X\cup Y[/tex], with [tex]X, Y[/tex] disjoint and open then [tex]\mathrm{cl}(M)[/tex] is contained in either [tex]X[/tex] or [tex]Y[/tex].

    Can you go from here?
  4. May 13, 2008 #3


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    Can you solve the problem if the closure has one additional point?
    How about two additional points?
  5. May 13, 2008 #4


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    the following version of connectedness makes all possible problems trivial:

    a set is connected iff all continuous maps to the set {0,1} are constant.
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