Understanding the disk model of projective space $\mathbb{P}^n(\mathbb{R})$ I'm working on Exercise 2.3.3 in Riemann Surfaces and Algebraic Curves: A First Course in Hurwitz Theory by Cavalieri and Miles:

In a previous definition, we realized $\mathbb{P}^n(\mathbb{R})$ as an identification/orbit space: let $\mathbb{R}^\ast = \mathbb{R} - \{0\}$ act on $\mathbb{R}^{n+1}$ by component-wise multiplication: $\lambda \cdot (X_0, X_1, \dots, X_n) = (\lambda X_0, \lambda X_1, \dots, \lambda X_N).$ Then $$\mathbb{P}^n(\mathbb{R}) = (\mathbb{R}^{n+1} - \{0\}) / \mathbb{R}^\ast.$$ We now present two alternative models for $\mathbb{P}^n(\mathbb{R})$ as an identification space, and leave it as an exercise that they yield homeomorphic results.
Sphere quotient. Consider the $n$-dimensional unit sphere $S^n \subset \mathbb{R}^{n+1}.$ The multiplicative cyclic group $\mu_2 = \{1,−1\}$ acts on the sphere by $$\pm 1 · (X_0, X_1,\dots, X_n) = (\pm X_0,\pm X_1,..., \pm X_n).$$ Then $\mathbb{P}^n(R)$ is the quotient space $S^n/\mu_2.$
Disk model. Consider the $n$-dimensional closed unit disk $\bar{D}^n \subset \mathbb{R}^n,$ and consider the antipodal equivalence relation on the points of its boundary: $x ∼ −x $ if and only if $||x|| = 1.$ Then $\mathbb{P}^n(\mathbb{R})$ is the identiﬁcation space $\bar{D}^n / ∼.$

I can understand the sphere quotient easily enough, but I can't seem to wrap my head around the disk model, and I can't tackle this problem very well until I can do that. Does anyone have a good way of picturing and understanding this model? I can't seem to find a good resource explaining this anywhere.
 A: The upper hemisphere, i. e. the subset $S^n_{+} \subset \mathbb{R}^n \times \mathbb{R}_{\geq 0}$ of the embedded sphere $S^n \subset \mathbb{R}^{n+1}$ is homeomorphic to the unit disc (it is the graph of the function $x \mapsto \sqrt{1 - |x|^2}$). Since every point in $\mathbb{P}^n(\mathbb{R})$ has a representative in the upper hemisphere, the map $S^n_+ / \mu_2 \to \mathbb{P}^n(\mathbb{R})$ is bijective and hence a homeomorphism (since its domain is compact and the target Hausdorff).
Here (by abuse of notation) the quotient by $\mu_2$ means that you quotient by the restriction of the equivalence relation you used for the sphere quotient. On the upper hemisphere however most points do not have an antipodal point. Actually  only those on the boundary (the equator) do. Hence the restricted equivalence relation consists only of identifying opposite points on the boundary. If push all of this down to $\mathbb{R}^n \times \{0\}$ (remember that this is a homeomorphism), you get the disc model described in your book.
