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This is a question from an interview. I am confused about this problem. I said yes in that interview because I remember something about Hilbert curve(I mean, is there a line that can fill a square completely?), but I am not sure. Is it right?

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    $\begingroup$ It was the second best answer that could be given. $\endgroup$ – Gae. S. Sep 20 '19 at 6:06
  • $\begingroup$ You probably mean a Peano curve? Those are not 1-1, only onto, so no homeomorphism. $\endgroup$ – Henno Brandsma Sep 20 '19 at 8:08
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An homeomorphism between a line and a disk can’t exist as a line minus a point is disconnected while a disk minus a point is connected.

However it exists surjective continuous maps between a line an a disk.

For an example of a curve filling a square, you can have a look at Lebesgue’s curve. Hilbert curve is indeed another example.

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Did you mean homeomorphism? Homomorphism doesn't make sense here.

There is no homeomorphism from a line onto a disk because the line becomes disconnected when you remove one point and that doesn't happen in disk.

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No.

Let a line $L$ be homeomorphic to a disk $B(x,R)$

Then excluding a point $y \in L$ we will have that $B(x,R) \setminus \{point\} $ will be homeomorphic to $L\setminus \{y\}$.

This cannot be true because then line minus a point is disconnected and the disk minus a point is still connected.

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