# Homeomorphism from (0,1) to R

1. Jun 6, 2012

### GridironCPJ

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. Jun 6, 2012

### 6.28318531

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
3. Feb 11, 2013

### Bachelier

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?