# I need help constructing an explicit bijection.

• Apr 11th 2012, 08:39 AM
allstar2
I need help constructing an explicit bijection.
Construct an explicit bijection g: [0,1] --> (0,1).
Hint: One way to do it would be three parts. One for g(0), one for g(1/n) for n belongs to N and one for everything else.
• Apr 11th 2012, 09:35 AM
emakarov
Re: I need help constructing an explicit bijection.
Does the following picture give you any ideas?

• Apr 11th 2012, 01:35 PM
allstar2
Re: I need help constructing an explicit bijection.
A little bit. I'm still not sure how to write it.
• Apr 11th 2012, 01:43 PM
Plato
Re: I need help constructing an explicit bijection.
Quote:

Originally Posted by allstar2
A little bit. I'm still not sure how to write it.

Define $f(0)=\frac{1}{2}$, $n\in\mathbb{Z}^+$ define $f(n^{-1})=(n+2)^{-1}$ and elsewise $f(x)=x$.
• Apr 11th 2012, 03:49 PM
allstar2
Re: I need help constructing an explicit bijection.
Thank you!
• Apr 17th 2012, 10:34 AM
HallsofIvy
Re: I need help constructing an explicit bijection.
A pretty standard example is this: the set of all rational numbers in (0, 1) is countable so can be "listed" $\{r_1, r_2, ..., r_n, ...\}$. Map 0 to $r_1$, 1 to $r_2$, and for all rational numbers $a_i$ to $a_{i+2}$. All irrational numbers in (0, 1) are mapped to themselves.