Is there a line homomorphism to a disk? 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?

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.
