Thread: I need help constructing an explicit bijection.

1. 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.

2. Re: I need help constructing an explicit bijection.

Does the following picture give you any ideas?

3. Re: I need help constructing an explicit bijection.

A little bit. I'm still not sure how to write it.

4. Re: I need help constructing an explicit bijection.

Originally Posted by allstar2
A little bit. I'm still not sure how to write it.
Define $\displaystyle f(0)=\frac{1}{2}$, $\displaystyle n\in\mathbb{Z}^+$ define $\displaystyle f(n^{-1})=(n+2)^{-1}$ and elsewise $\displaystyle f(x)=x$.

Thank you!

6. 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" $\displaystyle \{r_1, r_2, ..., r_n, ...\}$. Map 0 to $\displaystyle r_1$, 1 to $\displaystyle r_2$, and for all rational numbers $\displaystyle a_i$ to $\displaystyle a_{i+2}$. All irrational numbers in (0, 1) are mapped to themselves.