Let be the equivalence relation on R 2 that is defined as: For two points (x1, y1) and (x2, y2) in R 2 , we write (x1, y1) (x2, y2) if and only if there is an integer k such that x1 x2 = k, y1 = (1)k y2.

Let S = [0, 1) R be the half-closed-half-open strip as a subset of R 2 . Show that for any point P R 2 , there is a unique point P 0 S such that P 0 P (in other words, P 0 is a representative of the class cl(P)). Remark: You may think about how portals are built on the boundary of the closed strip [0, 1] R to describe the quotient set R 2/ as a geometric object.