ca.analysis and odes - Are there space filling curves for the Hilbert cube ?

Well there is indeed a "simple" construction of such a space filling curve.

Let $gamma:[0;1]rightarrow [0;1]^2$ be a space filling curve. Then one can obtain a space filling curve for $[0;1]^3$ by postcomposing with $id_mathbb{R}times gamma$. Then one can postcompose with $id_{mathbb{R}^2}times gamma$ and so on. Note that the first coordinates didn't change in the last step.

Putting all this together we get a map

$f:[0;1]rightarrow [0;1]^mathbb{N} qquad tmapsto (pr_1circ gamma circ (pr_2circ gamma)^{n-1}(t))_{nin mathbb{N}}, $
where $pr_i$ denote the obvious projections. This map can be seen as the infinite composition of the maps above. By the definition of the product topology this map is continuous.

Especially if we postcompose $f$ with the projection on the first $n$ coordinates, we just get a space filling curve (see above). Let us show, that a arbitrary element $x=(x_i)_{iin mathbb{N}}in [0;1]^mathbb{N}$ lies in the Image of $f$. We already know, that for each $n$ there is a element $y^n$ in Im$(f)$ agreeing with $x$ in the first $n$ coordinates.

As $[0;1]$ is compact, Im$(f)$ is compact and hence closed ($[0;1]^mathbb{N}$ is Hausdorff).

And $lim_{ntoinfty}y^n=x$. Hence $xin$ Im$(f)$. So $f$ is a continuous surjective map $[0;1]rightarrow [0;1]^mathbb{N}$.