/sci/ - Science & Math » Thread #14140408

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Anonymous Sun 23 Jan 22:20:37 2022 No.14140408 View Reply Original Report

Quoted By:

> there exists a bijection between
(0,1) and [0,1]

Anonymous

Anonymous Sun 23 Jan 2022 22:27:17 No.14140435 Report

Quoted By: >>14140482 >>14141285

I wonder how many people on this board can construct one

Anonymous

Anonymous Sun 23 Jan 2022 22:43:39 No.14140482 Report

Quoted By:

>>14140435
Easy, send 0 to 1/3, send 1 to 2/3. Then send 1/3 to 1/4 and 2/3 to 3/4. You can define this assignment iteratively, then take care of the remaining stuff that doesn't arise from this assignment by sending to itself.

Anonymous

Anonymous Mon 24 Jan 2022 02:33:14 No.14141199 Report

Quoted By:

https://planetmath.org/bijectionbetweenclosedandopeninterval

spooky

Anonymous

Anonymous Mon 24 Jan 2022 03:04:04 No.14141285 Report

Quoted By:

>>14140435
Pick any invertible map $f : \mathbb{N} \to [0,1] \cap \mathbb{Q}$ such that $f(0) = 0$ and $f(1) = 1$ .
Define $g : [0,1] \to (0,1)$ by
$ g(x) = f(f^{-1}(x)+2) \ \text{if $x \in \mathbb{Q}$ } \\ g(x) = x \ \text{otherwise} $
$g$ is clearly invertible by
$ g^{-1}(x) = f(f^{-1}(x)-2) \ \text{if $x \in \mathbb{Q}$ } \\ g^{-1}(x) = x \ \text{otherwise} $
hence g is a bijection.

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