writing a homeomorphism between these two spaces..! i 've tried to write a homeomorphism between below spaces but i've failed.
Could anyone tell how i should think for this?
$X = [-2,2]\times[-2,2] \setminus (\textrm{open disk})$     and     $Y = (\textrm{Disk with }r=2) \setminus (-1,1)\times(-1,1)$
 A: If you meant to remove $(-1,1)\times(-1,1)$ to get $Y$, then I recommend proceeding radially. The intersection of $X$ with any ray from the origin will be a closed interval, and the same is true of $Y.$ Use a nice consistent map in each case to transform said interval from the $X$-version to the $Y$-version, and you should get what you need.
A: You can use maps of the form $x/||x||$ to map both of them to the annulus. If the maps are $\phi$ and $\psi$, then the map you want is $\phi \circ \psi^{-1}$.
Edit: Rotate the figure consisting of the circle (of radius $2$) outside and square of side $2$ inside by an angle of $45^{\circ}$ counterclockwise. Now we will try to give a homeomorphism between this (call this F) and the annulus. Consider only the parts of these figures within the first quadrant. Then F is parametrized by $$((t \mbox{cos } \theta)^{2/t}, (t \mbox{sin } \theta)^{2/t})\mbox{,      }  t \in [1, 2]$$
Now the map $x \rightarrow t\frac{x}{||x||}$ gives the required homeomorphism. 
