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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2It was the second best answer that could be given. – Sep 20 '19 at 06:06
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You probably mean a Peano curve? Those are not 1-1, only onto, so no homeomorphism. – Henno Brandsma Sep 20 '19 at 08:08
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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.
mathcounterexamples.net
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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.
Kavi Rama Murthy
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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.
Marios Gretsas
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