 Solution

ID: past-exam-of-the-topology-course-of-the-university-of-oxford/2021/2/a/i/solution

S^{2} = {u \in R^{3} : ∥ u ∥ = 1}, u \sim_{S} v ⟺ v = u or v = - u .

Then

RP^{2} = S^{2} / \sim_{S}

D^{2} = {(x, y) \in R^{2} : x^{2} + y^{2} \leq 1}, u \sim_{D} v ⟺ u = v or (u, v \in \partial D^{2} and v = - u) .

Give

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.

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