Past exam of the topology course of the University of Oxford / 2021 / 2 / a / i / Solution

For the sphere model of the real projective plane, letThen ${\mathbb{RP}}^2=S^2/{\sim }_S$ with the quotient topology: each equivalence class is an antipodal pair.

For the closed disc model, letGive $D^2/{\sim }_D$ the quotient topology. Only antipodal pairs on the boundary circle are identified in the closed disc model; interior points remain distinct.