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Can you give me an example of a bijective function $f:\mathbb{C}\rightarrow\mathbb{R}$? Can you parameterize a continuous plane with a continuous line?

ty.
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  • For the bijection, look here: https://mathoverflow.net/questions/126069/bijection-from-mathbbr-to-mathbbr2 – Lee Mosher Jul 11 '17 at 18:11
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    For the parameterization, do you mean a continuous surjection? or a continuous bijection? or what do you mean? – Lee Mosher Jul 11 '17 at 18:11

3 Answers3

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Regarding the "continuous" part of the question, it's unclear what you want, but there isn't much good news.

There does not exist a continuous injection $\mathbb R^2\to\mathbb R$. Any such function would restrict to a continuous injection from the circle $S^1\to\mathbb R$, which is already impossible by the intermediate value theorem.

Also, there does not exist a continuous bijection $\mathbb R\to\mathbb R^2$. This is less obvious. See this question: Is there a continuous bijection from $\mathbb{R}$ to $\mathbb{R}^2$

Chris Culter
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  • I suppose I could be more clear: Say I have a Vectorspace $\mathbb{R^2}$ over the field $\mathbb{R}$ and I want to find a bijective function $g:\mathbb{R^2}\rightarrow\mathbb{R}$ where for all $x_0\in \mathbb{R^2}\quad \lim_{x\rightarrow x_0} f(x)=f(x_0)$. – ty. Jul 11 '17 at 18:36
  • Thanks for the link – ty. Jul 11 '17 at 18:42
  • @ty. Ah, so you want a continuous bijection $\mathbb R^2\to\mathbb R$. Unfortunately, that's impossible; see my second paragraph. – Chris Culter Jul 11 '17 at 18:50
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Since you tagged this as linear algebra, let's proceed as follows:

Let $\{\alpha_j\}_{j\in J}$ be a basis of the $\Bbb Q$-vector space $\Bbb R$, then $\{u\alpha_j\}_{(j,u)\in J\times\{1,i\}}$ is a basis of $\Bbb C$. Since $J$ is infinite, it has the same cardinality as $J\times\{1,i\}$. Any bijection $J\to J\times\{1,i\}$ then gives rise to a $\Bbb Q$-linear map $\Bbb R\to \Bbb C$.

Hagen von Eitzen
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Yes there is a way. You can use a Hilbert Curve to map from a finite (bounded) 2D square area of size (L * L) onto a 1D line of length (L * L), however, in a separate step, the shape and size of the square can be transformed with an afine transformation function and the line can be scaled to any size. The beauty of this approach is that each iteration of the Hilbert curve will quickly converge on a ever more accurate mapping, so any degree of accuracy is possible. Additionally, this mapping is reversible to any desired resolution. This is continuous from the point of view that there are no gaps in the mapping, but this is not continuous from the point of view of 2D motion or the introduction of a raster "resolution".

Will
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