# Peano Space filling Curve Surjective

I'm having a little trouble understanding why the peano space filling curve is surjcetive. If we have the usual Hilbert Peano space filling curve $f:[0,1]\rightarrow [0,1]^2$ and we know that $f$ is continuous. Do we have to use the compactness of $[0,1]$?

So that the image of $f$ is compact in $\mathbb{R}^2$ and then it is closed. Once we have it is closed and then by the construction of the function $f$ we have that each point in $[0,1]^2$ is a limit point of $f$? so then it is in $f([0,1])$? Is this the right idea?

Thanks for any help

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All the work goes into constructing a continuous function $f:\ [0,1]\to[0,1]^2$ with a dense image. The rest is "abstract blabla".