sphere-filling curve

Let $S^2$ denote the $2$-dim sphere in $\mathbb R^3$. I am interested in finding a space-filling curve, i.e. a map $\varphi: [0,1]\to S^2$ that is continuous and onto.

We know that there is such a space-filling curve onto $[0,1]^2$ from Peano's and Hilbert's results.

Know my idea was to consider a unit cube in $\mathbb R^3$. Then I want to take a path that traverses enough edges of the cube (or just take a Hamiltonian path). From an edge I want to fill its face with Peano's curve and then go back to the edge after filling. This construction yields a cube-filling curve. Then I blow the cube up to the sphere and I am done.

However, this seems to simple to me. Does that work or do I miss something?

Solution 1:

Your construction seems doable. Here's an alternate rough sketch of a construction (I hope it's ok if I don't produce the details):

Choose space filling curves $f_{m, n} : [0, 1] \to [n, n+1] \times [m, m + 1]$ for each integer $m, n$ so that they glue to a continuous surjection $f : \Bbb R \to \Bbb R^2$ in the sense that for each unit interval $I_k = [k, k +1] \subset \Bbb R$ with $k$ an integer, $f|_{I_k}$ is $f_{m, n}$ for some choice of $m, n$ (depending on $k$). That is, glue a bunch of space filling curves on each $[n, n + 1] \times [m, m + 1]$ systematically so that the endpoints agree. Continuity of $f$ is guaranteed by gluing lemma since endpoints of $f_{m, n}$ agree.

Note that the above construction can be done because you can always choose a space filling curve $f : I \to I^2$ so that the endpoint $f(0)$ and $f(1)$ are whatever you want them to be (you can cook up a Peano or Hilbert like construction with endpoints given).

Staring a bit carefully tells you that you can actually choose the endpoints so that $f$ is proper (that is, glue $f_{m, n}$ in a way so that $\{f(k)\}$ heads off to infinity as $k \to +\infty$ or $k \to -\infty$). Once that is done, proper maps can be extended to one-point compactifications, so $f$ extends to a space filling curve $\tilde{f} : S^1\to S^2$. Now compose with the map $g : I \to S^1$, $g(x) = e^{2\pi ix}$ and you're done.