Editing Space-filling curve (section)

== Outline of the construction of a space-filling curve ==
Let <math>\mathcal{C}</math> denote the [[Cantor space]] <math>\mathbf{2}^\mathbb{N}</math>.

We start with a continuous function <math>h</math> from the Cantor space <math>\mathcal{C}</math> onto the entire unit interval <math>[0,\, 1]</math>.  (The restriction of the [[Cantor function]] to the [[Cantor set]] is an example of such a function.)  From it, we get a continuous function <math>H</math> from the topological product <math>\mathcal{C} \;\times\; \mathcal{C}</math> onto the entire unit square <math>[0,\, 1] \;\times\; [0,\, 1]</math> by setting

<math display="block">H(x,y) = (h(x), h(y)). \, </math>

Since the Cantor set <math>\mathcal{C}</math> is [[homeomorphic]] to its cartesian product with itself <math>\mathcal{C} \times \mathcal{C}</math>, there is a continuous bijection <math>g</math> from the Cantor set onto <math>\mathcal{C} \;\times\; \mathcal{C}</math>. The composition <math>f</math> of <math>H</math> and <math>g</math> is a continuous function mapping the Cantor set onto the entire unit square.  (Alternatively, we could use the theorem that every [[compact set|compact]] metric space is a continuous image of the Cantor set to get the function <math>f</math>.)

Finally, one can extend <math>f</math> to a continuous function <math>F</math> whose domain is the entire unit interval <math>[0,\, 1]</math>.  This can be done either by using the [[Tietze extension theorem]] on each of the components of <math>f</math>, or by simply extending <math>f</math> "linearly" (that is, on each of the deleted open interval <math>(a,\, b)</math> in the construction of the Cantor set, we define the extension part of <math>F</math> on <math>(a,\, b)</math> to be the line segment within the unit square joining the values <math>f(a)</math> and <math>f(b)</math>).