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Homeomorphism from (0,1) to R

  1. Jun 6, 2012 #1
    1. The problem statement, all variables and given/known data
    Find an explicit homeomorphism from (0,1) to R.


    2. Relevant equations
    A homeomorphism from (-1,1) to R is f(x)=tan(pi*x/2).


    3. The attempt at a solution
    I'm horrible a modifying trig functions. Obviously, to shift by b you add b to (x) and you can change the frequency by including a scalar to pi. I tried f(x)=tax(2pi*(x+1)/2), but this doesn't do the trick.
     
  2. jcsd
  3. Jun 6, 2012 #2
    You are close, you just need to change a few things. Also Do you have to use a trig function? What about x/1-x2, that maps (-1,1) -> ℝ, could you modify that?
     
    Last edited: Jun 6, 2012
  4. Feb 11, 2013 #3
    Consider: ##f: (0,1) → \mathbb{R} \\
    \ \ x → \frac{2x-1}{1-(2x-1)^2} ##

    This mapping is a Homeomorphism. meaning a Bijection.
    could someone specify a metric on (0,1) that defines (the same topology) as the abs. value (i.e. the usual) metric and makes this open interval into a complete set?
     
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