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.
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($\displaystyle \pi$x - $\displaystyle \pi$/2)
You should not be confused: the real number set is closed as a topological space.
There is a bijection $\displaystyle \left[ {0,1} \right]\mathop \Leftrightarrow \limits^f \left( {0,1} \right)$, this done by shifting a countable set.
$\displaystyle g(x)=\tan\left(x\pi-\frac{\pi}{2}\right)$ is a bijection $\displaystyle \mathbb{R} \Leftrightarrow \left( {0,1} \right)$.
What about a composition of those two?
See also this thread for a bijection between [0, 1] and (0, 1).