Bijection between $\mathbb{R}^n$ and $\mathbb{R}^m$

Question

I've heard that it is possible to present a bijection $\phi : \mathbb{R}^n \to \mathbb{R}^m$ due to Cantor that although not continuous show that those sets have the same cardinality independent of the dimension as a vector space. The problem is that I couldn't find about it anywhere. Can someone tell me where I can find about this bijection and how to show it's existance?

Thanks very much in advance!

You will find it interesting to read about space-filling curves. — Lord_Farin, Aug 22 '13 at 11:15
@Lord_Farin: while related and interesting, space-filling curves are not bijections. — Anthony Carapetis, Aug 22 '13 at 11:16
@Anthony Thanks for pointing that out; I'm not sure how that mistake ended up in my head. :blush: — Lord_Farin, Aug 22 '13 at 11:22
You could use this $\mathbb{R} \to \mathbb{R}^m$ bijection and its inverse $\mathbb{R}^n \to \mathbb{R}$, composing them would yield $\mathbb{R}^n \to \mathbb{R}^m$ bijection. However, having a bijection is not that simple, maybe two injections would be enough for you? — dtldarek, Aug 22 '13 at 11:35
Hi @dtldarek, thanks for the answer! I'm really trying to find a bijection, because I've always heard it is possible and never saw it done. I think that your idea will work very well. Thanks for your help! — Gold, Aug 22 '13 at 11:38

score 0 · Answer 1 · answered Aug 22 '13 at 11:26

First, notice that it suffices to find a bijection $\psi : \mathbb{R} \to \mathbb{R^2}$ - once we have this, we can easily extend this by (for $n<m$)

$$ \phi(x_1, \ldots, x_n) = (\psi(x_1), \ldots \psi(x_{m-n}), x_{m-n+1}, \ldots, x_n).$$

The standard example of $\psi$ is to first biject $\mathbb{R}$ with $[0,1)$ (usually via the tangent function and some trickery to get 0 in there) and then interleave the decimal expansion - e.g. send $0.123456\ldots$ to $(0.135\ldots,0.246\ldots)$. There's some issue with the non-uniqueness of decimal expansions, but it can be gotten around with some modification. I'm having trouble finding a reference for this but there have been a lot of posts on here about bijections $\mathbb{R} \to \mathbb{R}^2$; try searching and reading a few.

Bijection between $\mathbb{R}^n$ and $\mathbb{R}^m$

1 Answers1

Linked

Related