Space-filling curve

Example

A space-filling curve is a continuous surjection from $[0, 1]$ onto the square $[0, 1]^2$. Here is one, due to Lebesgue. Each point $x$ of the Cantor set $C$ has a unique expansion $x = \sum_{n \ge 1} 2 a_n 3^{-n}$ with digits $a_n \in \set{0, 1}$. Set $$f(x) = \left( \sum_{n \ge 1} a_{2n-1} 2^{-n}, \ \sum_{n \ge 1} a_{2n} 2^{-n} \right).$$ On each open interval removed in the construction of $C$, extend $f$ linearly between its values at the two ends.

The check has three steps. The map $f$ is onto. Take binary expansions of both coordinates of a point of $[0, 1]^2$; interleaving their digits gives a point of $C$ that $f$ sends to it. The map $f$ is continuous on $C$. Two points of $C$ less than $3^{-2m}$ apart share their first $2m$ digits, so their images differ by at most $2^{-m}$ in each coordinate. The two ends of a removed interval of length $3^{-k}$ share their first $k - 1$ digits. So short removed intervals carry small linear pieces, and the extended map is continuous.