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Topology (complex analysis)

  1. Feb 12, 2013 #1
    1. The problem statement, all variables and given/known data
    Let S={zεℂ: |z|<1 or |z-2|<1}. show that S is not connected.


    2. Relevant equations
    My prof use this definition of disconnected set.

    Disconnected set - A set S [itex]\subseteqℂ[/itex] is disconnected if S is a union of two disjoint sets A and A' s.t. there exists open sets B and B' with A [itex]\subseteq B[/itex]
    and A' [itex]\subseteq B'[/itex].

    3. The attempt at a solution

    So here's my solution. I let A={zεℂ: |z|<1} and A'={zεℂ: |z-2|<1}. in order for me to show that S is disconnected set, i need to show the following.

    i.) S=A U A'
    ii.) A and A' are disjoint
    iii.) A and A' are open sets.

    Am i doing the right way? Thanks
     
  2. jcsd
  3. Feb 12, 2013 #2

    haruspex

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    That doesn't work. is it perhaps: A set S [itex]\subseteqℂ[/itex] is disconnected if S is a union of two sets A and A' s.t. there exists disjoint open sets B and B' with A [itex]\subseteq B[/itex] and A' [itex]\subseteq B'[/itex]?
    Looks good to me.
     
  4. Feb 12, 2013 #3
    sorry, you're right, i forgot the "disjoint" word.


    is there any other way (probably, easier) to prove that S is a disconnected set?
     
  5. Feb 12, 2013 #4

    Dick

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    You are doing fine. You just have the three (easy!) things to prove about A and A'. And you may not have to do a terribly formal proof of each. Just indicate why they are true.
     
    Last edited: Feb 12, 2013
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