I have to give an equivalence relation on real projective plane $\mathbb{P}^2$ such that $\mathbb{P}^2/\sim$ is homeomorphic to $S^2$, the two-dimensional sphere.
Once I have the equivalence relation I know how to prove that it is indeed homeomorphic. However I don't know how to find such an equivalence relation. Can you give me a hint?
Not sure if it works, but you may consider the following equivalence relation. Let $u_1:=(x_1:y_1:z_1), \ u_2:=(x_2:y_2:z_2)\in \mathbb P^2$, and define $u_1 \sim u_2$ if and only if $u_1=u_2$ or $z_1=z_2=0$. Geometrically you are identifying all the directions contained in the $xy$-plane, which should represent the "south pole"; the other points are not identified and represents the other points of the sphere, e.g. $(0:0:1)$ will be the "north pole".