Relationship b/w Connectedness and Homeomorphisms

  1. I'm using Real Mathematical Analysis by Pugh to supplement my analysis class, and the book has been clear thus far, but I've been stuck for days on a concept I've had a hard time understanding.

    Just for reference, here is how a homeomorphism is defined:

    And here is how connectedness is defined:

    I understand that M connected and M homeomorphic to N implies N connected, and that M connected, f:M->N continuous, and f onto implies N connected. However, what I don't understand are examples such as the following:

    What I don't understand is how M connected and N disconnected implies M, N not homeomorphic, as in the first example. If I take this as being true, I understand the logic of the second example, with the detailed explanation of the process of removing a point. However, I still don't see M connected and N disconnected implying M,N not homeomorphic.

    Many thanks to anyone who can help me out.
     
    Last edited: Apr 2, 2011
  2. jcsd
  3. lavinia

    lavinia 2,056
    Science Advisor

    homeomorphisms preserve open sets.
     
  4. Office_Shredder

    Office_Shredder 4,499
    Staff Emeritus
    Science Advisor
    Gold Member

    So if N is not connected, and a homeomorphism exists from M to N when M is connected, we get a contradiction (because N is connected and not connected at the same time)
     
  5. Your understanding of one should imply your understanding of the other!
     
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