I've been looking at equivalence relations on the unit square: $[0,1] \times [0,1]$ that give rise to various surfaces such as the m$\ddot{\mathrm{o}}$bius strip, but I'm not too sure about the Klein bottle and real projective two-space?
My only thoughts regarding the real projective two-space are $x\sim y$ if $x=y$ or $(x,0)\sim (y,1)$ and $(0,x)\sim (1,y)$, and I don't have a clue about the Klein bottle
The Klein bottle is obtained from a cylinder side by gluing the end-circles together in opposite orientation (so you don't get a torus!).
What you said about projective two-space is wrong, when $\sim$ is a relation on the unit square and you write $x\sim y$, $x$ and y are elements of the unit square, so $(x,0)$ would be something like a triple. What is it you intended to say?