Outline of The Construction of A Space-filling Curve
Let denote the Cantor space .
We start with a continuous function from the Cantor space onto the entire unit interval . (The restriction of the Cantor function to the Cantor set is an example of such a function.) From it, we get a continuous function from the topological product onto the entire unit square by setting
Since the Cantor set is homeomorphic to the product, there is a continuous bijection from the Cantor set onto . The composition of and is a continuous function mapping the Cantor set onto the entire unit square. (Alternatively, we could use the theorem that every compact metric space is a continuous image of the Cantor set to get the function .)
Finally, one can extend to a continuous function whose domain is the entire unit interval . This can be done either by using the Tietze extension theorem on each of the components of, or by simply extending "linearly" (that is, on each of the deleted open interval in the construction of the Cantor set, we define the extension part of on to be the line segment within the unit square joining the values and ).
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