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Math Help - Homeomorphism

  1. #1
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    Homeomorphism

    How do I prove that two closed intervals,
    [a_1,b_1]=\{x \in  \Re : a_1 \le x \le b_1\} \ \& \ [a_2,b_2] \  (a_i  <b_i) are homeomorphic? Prove by writing down the formula for a homeomorphism.
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  2. #2
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    I didn't say enything
    Last edited by ialbrekht; October 4th 2009 at 07:39 AM.
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  3. #3
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    Quote Originally Posted by bigdoggy View Post
    How do I prove that two closed intervals,
    [a_1,b_1]=\{x \in  \Re : a_1 \le x \le b_1\} \ \& \ [a_2,b_2] \  (a_i  <b_i) are homeomorphic? Prove by writing down the formula for a homeomorphism.
    Try f(x) = \left[ {\frac{{b_2  - a_2 }}{{b_1  - a_1 }}} \right]\left( {x - a_1 } \right) + a_2
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  4. #4
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    Quote Originally Posted by Plato View Post
    Try f(x) = \left[ {\frac{{b_2  - a_2 }}{{b_1  - a_1 }}} \right]\left( {x - a_1 } \right) + a_2
    How do you arrive at that!?
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  5. #5
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    To prove the homeomorphism you need to find function f:A->B with following properties Homomorphism - Wikipedia, the free encyclopedia.

    In your case you have 2 intervals [a1,b1] & [a2,b2]. You can make such function by stretching one interval and then move it to the beginning of the second one.

    -------[a1............b1]----------
    ------------[a2...........b2]------
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  6. #6
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    Quote Originally Posted by bigdoggy View Post
    How do you arrive at that!?
    Well I certainly hope that you recognize that as the equation of a line.
    Linear functions are continuous bijections.
    This particular line is determined by the pairs (a_1,a_2)~\&~(b_1,b_2)
    It maps [a_1,b_1]\mapsto[a_2,b_2].
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