I am looking at space filling curves. Essentially their is a mapping $f: I \to \mathcal{Q}$ where I is an interval in $\mathbb{R}$ such as $[0,1]$ and $\mathcal{Q}$ is a square $[0,1]^2$.

For the mapping $f$ to define a space filling curve it has to be continuous, surjective and we have to show every $t \in I$ is uniquely mapped to a point $f(t) \in \mathcal{Q}$

What does it mean to say uniquely mapped?

I thought it meant every point in the pre image had a unique point in the image. But this would make the mapping injective but a space filling curve can not be injective.

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    $\begingroup$ It is simply means that $f$ is a function. $\endgroup$ – Crostul Jul 20 '16 at 13:40
  • $\begingroup$ @crostul how do you prove this? $\endgroup$ – Al jabra Jul 20 '16 at 13:52
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    $\begingroup$ I would use the expression “$p$ is uniquely mapped to $q$” only when the function involved some choices along the way, and it would be incumbent on me to verify that the choices did not affect the result. Beyond that, I agree with @Crostul, that the expression adds nothing to the mathematical discussion: “$p$ is mapped to $q$” would usually suffice. $\endgroup$ – Lubin Jul 20 '16 at 14:35

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