How construct homeomorphism from $R^2$-{$(x,y)$ : $x,y$ : integers } to $R^2$-{$(x,y)$ : there are integers n,m such that $(x-n)^2+(y-m)^2<1/10$}?
This is from 'Introduction to topology', Gamelin. I read his solution but I can't understand. Here is his solution.
'First construct a homeomorphism of a closed disk minus the closed disk with same center and half the radius, onto the same (full size) closed disk minus the center, with the additional property that it is constant on the boundary (e.g., for the closed unit disk, $(r,\theta) \rightarrow (2r-1,\theta)$. Then apply this to appropriate kisks centered at the integral lattice points in the plane.