= Solution
{c}
For the <sphere> model of the <real projective plane>, let
$$
S^2=\{u\in\mathbb R^3:\|u\|=1\},\qquad u\sim_S v\Longleftrightarrow v=u\text{ or }v=-u.
$$
Then $\mathbb{RP}^2=S^2/{\sim_S}$ with the <quotient topology>: each <equivalence class> is an <antipodal pair>.
For the <closed disc> model, let
$$
D^2=\{(x,y)\in\mathbb R^2:x^2+y^2\leq1\},\qquad u\sim_D v\Longleftrightarrow u=v\text{ or }(u,v\in\partial D^2\text{ and }v=-u).
$$
Give $D^2/{\sim_D}$ the <quotient topology>. \b[Only <antipodal pairs> on the <boundary> <circle> are identified in the <closed disc> model; interior points remain distinct.]
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