# Is there a line homomorphism to a disk? [duplicate]

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?

• It was the second best answer that could be given. – Gae. S. Sep 20 '19 at 6:06
• You probably mean a Peano curve? Those are not 1-1, only onto, so no homeomorphism. – Henno Brandsma Sep 20 '19 at 8:08

## 3 Answers

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.

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.

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.