Homeomorphisms of quotient spaces. (With pictures)

Question

I have a question about homeomorhic spaces: $\mathbb{R}^2$; $\mathbb{R}^2 /I$; $\mathbb{R}^2/D^2$; $\mathbb{R}^2/I^2$.

My attempt for first pair... I try to understand deformation of rectangular lattice via homeomorphism. Is my picture correct ($\mathbb{R}^2 \to \mathbb{R}^2 / I$)? How I can write expression for homeomorphism between $\mathbb{R}^2 \to \mathbb{R}^2 / I$ like $f(x,y)=\dots$?

Is it correct? Thanks!