Topology question

  • Thread starter metalbec
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  • #1
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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.

Thanks
 

Answers and Replies

  • #2
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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?
 
  • #3
NateTG
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Can you solve the problem if the closure has one additional point?
How about two additional points?
 
  • #4
mathwonk
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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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