Rank-one orthogonal projections form the projective space

Idea. A line through the origin is determined by a unit vector up to sign, and the orthogonal projection onto it is $xx^T$, which does not see the sign.

Step 1: description of $\mathcal P_1$. If $P\in\mathcal P_1$, its image is a line $\mathbb{R}x$ with $|x|=1$, and $P$ is the orthogonal projection onto it: $Py=(x\cdot y)x=xx^Ty$. So $P=xx^T$. Conversely, for a unit vector $x$, $xx^T$ is symmetric, satisfies $(xx^T)^2=x(x^Tx)x^T=xx^T$, and has rank $1$.

Step 2: the map. $\phi:S^{n-1}\to\mathcal P_1$, $\phi(x)=xx^T$, is continuous and onto by Step 1. Moreover $xx^T=yy^T$ iff $x=\pm y$: apply both sides to $x$ to get $x=(y\cdot x)y$, so $x$ is a multiple of $y$, and both are unit vectors.

Step 3: homeomorphism. $\phi$ induces a continuous bijection $\bar\phi:\mathbb{RP}^{n-1}=S^{n-1}/\{\pm1\}\to\mathcal P_1$. The source is compact (a quotient of the sphere) and $\mathcal P_1\subset M_n(\mathbb{R})$ is Hausdorff. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism.

Remark. The same construction with $x\in\mathbb{C}^n$ and $xx^*$ gives $\mathbb{CP}^{n-1}$ as the space of rank-one Hermitian projections. For $n=2$, $\mathcal P_1$ is a circle: $\mathbb{RP}^1\cong S^1$.

Lesson. Compact-to-Hausdorff continuous bijections are homeomorphisms: use them to identify quotient spaces.