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.

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- Apr 11th 2012, 08:39 AMallstar2I 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 AMemakarovRe: I need help constructing an explicit bijection.
Does the following picture give you any ideas?

https://lh4.googleusercontent.com/-K.../bijection.png - Apr 11th 2012, 01:35 PMallstar2Re: I need help constructing an explicit bijection.
A little bit. I'm still not sure how to write it.

- Apr 11th 2012, 01:43 PMPlatoRe: I need help constructing an explicit bijection.
- Apr 11th 2012, 03:49 PMallstar2Re: I need help constructing an explicit bijection.
Thank you!

- Apr 17th 2012, 10:34 AMHallsofIvyRe: 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.