There is a surjective continuous map $[0;1]\rightarrow [0;1]^2$ ("space filling curve"). Using such a map one can easily get space filling curves for all finite dimensional cubes. 
So my question is: Is there a space filling curve of the Hilbert cube 
$[0;1]\rightarrow [0;1]^\mathbb{N}$ ?



Are there space filling curves for the Hilbert cube?
Ah, it's called the "Hahn–Mazurkiewicz theorem".


<https://en.wikipedia.org/wiki/Peano_space>


And it apparently appears in the Willard text.


