1. ## Bijection

Give an explicit example of a bijection from [0,1] to the set of real numbers.

I'm confused on how to do this one because it is a closed set.

2. ## Re: Bijection

Recall that a finite composition of bijections is also a bijection. [I don't remember the infinite case]

Let f be a bijection between [0,1] and (0,1) and let g be a bijection between (0,1) and R. then g o f is a bijection between [0,1] and R

f(x): [0, 1] -> (0, 1) by

f(0) = 1/2
If n is a positive integer, then f(1/n) = 1/(n + 2)
Otherwise (that is, if x is not zero and cannot be written in the form 1/n for n a positive integer), then f(x) = x.

g(x) = tan( $\pi$x - $\pi$/2)

3. ## Re: Bijection

Originally Posted by lovesmath
Give an explicit example of a bijection from [0,1] to the set of real numbers. I'm confused on how to do this one because it is a closed set.
You should not be confused: the real number set is closed as a topological space.

There is a bijection $\left[ {0,1} \right]\mathop \Leftrightarrow \limits^f \left( {0,1} \right)$, this done by shifting a countable set.

$g(x)=\tan\left(x\pi-\frac{\pi}{2}\right)$ is a bijection $\mathbb{R} \Leftrightarrow \left( {0,1} \right)$.

What about a composition of those two?