I'm trying to construct a homeomorphism from $[0,1]\times[0,1]$ to $\overline{D}(0,1)$. I'm pretty sure there is one.
I've been trying to work geometrically : mapping $[0,1]\times[0,1]$ to $[-1/2,1/2]\times[-1/2,1/2]$ (which is realatively easy) and then mapping the boundary of the latter to the boundary of $\overline{D}(0,1)$ and generalizing to the inside of both... I think it could work but I can't construct the map, can you help me?
Thanks a lot!
Assuming you are considering these spaces as subspaces of $\mathbb R ^2$ with the Euclidean topology, here are some hints:
1) Given line segments $[0,a]$ and $[0,b]$, for some positive $a,b$, can you construct a particularly nice (i.e., linear) homeomorphism $[0,a]\to [0,b]$?
2) The closed disk is just a bunch of line segments glued together, just think of all of the radii.
3) The square $[0,1]\times [0,1]$ is just a bunch of line segment glued together, just think of all the line segments from the centre of the square to its boundary.
4) You should now be able to define a function which maps the centre of the closed disk to the centre of the square, and linearly maps each radius to a line segment of the square's centre to its boundary. Check that it is a homeomorphism.