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Math Help - Quick question... homeomorphisms...

  1. #1
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    Quick question... homeomorphisms...

    Can a homeomorphism exist between an open and a half open set?
    (ie: (0,1) and [0,1))

    I know that to be a homeomorphism, a bijection must exist, they must be continuous, and they must have a continuous inverse... Where to go from here?

    The fact that 0 is not included within the first set, would that alone make it so that it is not homeomorphic?

    Thank you.
    Jia Lin
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  2. #2
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    I don't think there is even a surjective continous function from (0,1) to [0,1) (I'm still trying to prove it though)
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  3. #3
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    I know what the answer should be... they definitely aren't homeomorphic, but it is an awkward proof.

    Thank you for your help.
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  4. #4
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    Nevermind, I just found a counterexample to my argument.
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  5. #5
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    Quote Originally Posted by Majialin View Post
    Can a homeomorphism exist between an open and a half open set?
    (ie: (0,1) and [0,1))

    I know that to be a homeomorphism, a bijection must exist, they must be continuous, and they must have a continuous inverse... Where to go from here?

    The fact that 0 is not included within the first set, would that alone make it so that it is not homeomorphic?

    Thank you.
    Jia Lin
    Let A = [0, 1) and B = (0, 1). Suppose there exists an homeomorphism f between A and B. Then f((0,1)) has to be a connected set, because (0,1) is a connected set and "connectedness" is a topological invariant. We see that A\{0} is a connected set, but B\{f(0)} is a disconnected set. Contradiction !

    Thus there exists no homeomorphism between A and B.
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  6. #6
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    Thank you very much!
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