I have some questions on the generation of the Hilbert's space-filling curve. Any help to clarify doubts a-e would be very appreciated.
The Hilbert's space-filling curve is a function $f_h:[0,1]\rightarrow [0,1]^2$ built following this procedure of 4 steps (e.g. from here):
(step 1) Assume that $[0,1]$ can be mapped continuously onto $[0,1]^2$.
(a) Could you clarify step 1? Does it mean that I should take a specific continuous function from $[0,1]$ to $[0,1]^2$? Why is it an assumption?
(b) It is not clear to me why I need to think about a continuous map: is continuity a necessary condition for having a partition as described at step 2?
(c) Is it true that if I change the initial continuous map from $[0,1]$ to $[0,1]^2$ then I will have a different initial partition at step 2? Does this imply that we can get different Hilbert space-filling curves following these steps?
(step 2) If we partition $[0,1]$ into 4 congruent subintervals, then it should be possible to partition $[0,1]^2$ into 4 congruent subsquares, such that each subinterval is continuously mapped onto one of the subsquares.
(step 3) The subintervals and subsquares should be such that: adjacent subintervals correspond to adjacent subsquares with an edge in common (it preserves continuity), and the inclusion relationships are preserved, i.e. if a square corresponds to an interval, then its subsquares corresponds to the subintervals of that interval (the mapping at the $n$-th iteration preserves the mapping at the $n-1$-th iteration).
(d): Are there several ways to make step (3) happening? Does this imply that we can get different Hilbert space-filling curves?
(step 4) Repeat the argument for each subinterval and subsquare ad infinitum.
(e) At a given step $n$, we don't know yet which is $f_h(t)$ for any $t\in [0,1]$. However, should I think about having already a surjective map from $[0,1]$ to $[0,1]^2$ or just a 1-1 association between subintervals and subsquares?
I can see how you would find the assumption that a space-filling curve exists confusing. After all, the point is to construct a space filling curve in the first place - why would we assume it can be done?
I think of it this way: If we do assume that there is a space filling curve $f_h:[0,1]\to [0,1]^2$, then it makes sense to suppose that $f_h$ might map the sub-intervals $[0,1/4]$, $[1/4,1/2]$, $[1/2,3/4]$, and $[3/4,1]$ onto sub-squares of $[0,1]^2$. This points us to think in terms of self-similarity and, ultimately, Hilbert's actual construction.
And, yes, there are a number of potential variations in the construction. One well known variant is Moore's curve.