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Relationship b/w Connectedness and Homeomorphisms

  1. Apr 2, 2011 #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. Apr 2, 2011 #2

    lavinia

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    homeomorphisms preserve open sets.
     
  4. Apr 2, 2011 #3

    Office_Shredder

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    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. Apr 3, 2011 #4
    Your understanding of one should imply your understanding of the other!
     
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