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Math Help - How to find homeomorphism between sets

  1. #1
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    How to find homeomorphism between sets

    Find homeomorphism between sets
    1 - \left[ {a,\infty } \right[\& \left[ { - 1,1} \right[
    2- \left] { - \infty ,1} \right]\& \left] { - 1,1} \right]\,<br />

    is the solution


    \begin{array}{l}<br />
 f_1 \left( x \right) = \frac{{x - 2a}}{x} \\ <br />
 f_2 \left( x \right) = \frac{{2 - x}}{x} \\ <br />
 \end{array}<br />
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  2. #2
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    Re: How to find homeomorphism between sets

    Quote Originally Posted by rqeeb View Post
    Find homeomorphism between sets
    1 - \left[ {a,\infty } \right[\& \left[ { - 1,1} \right[
    2- \left] { - \infty ,1} \right]\& \left] { - 1,1} \right]\,<br />

    is the solution


    \begin{array}{l} f_1 \left( x \right) = \frac{{x - 2a}}{x} \\ f_2 \left( x \right) = \frac{{2 - x}}{x} \\ \end{array}
    These are good attempts, but there is a snag. The solution for 1) is correct provided that a>0. But it does not work if a<0 because in that case x can take the value 0, and the function f_1(x) has a vertical asymptote there.

    The solution to 2) fails for the same reason it goes wrong at x=0. But you can easily put that right in this case by flipping the function over and defining f_2(x) = \tfrac x{2-x}. Since x does not take the value 2, the denominator is not going to vanish.

    TeX tip: the TeX engine used in this forum does not like line breaks. If you write your TeX input all on one line, as

    [TEX]\begin{array}{l} f_1 \left( x \right) = \frac{{x - 2a}}{x} \\ f_2 \left( x \right) = \frac{{2 - x}}{x} \\ \end{array}[/TEX] ,

    it will be rendered without the tiresome <br />
s.
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  3. #3
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    Re: How to find homeomorphism between sets

    yes .. u r right
    Last edited by rqeeb; September 29th 2011 at 07:33 PM.
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