Definition. Real Projective $n$-space as quotient of $S^n$ [math-UGKQ]

Another famous view is viewing $\mathbb{R}P^n$ as the quotient of the sphere $S^n$, because it’s obvious that each element (a line) of projective $n$-space intersects $S^n$ exactly two points, and the two points are antipodal points!

Use $l_a$ to represent an element of $\mathbb{R}P^n$ that intersects $S^n$ at $a$.

Therefore, if we make a quotient relation:

Then we can see that $\mathbb{R}P^n \simeq S^n/\sim$, this homeomorphism can be used to transfer the $C^r$-structure on $S^n$ to $\mathbb{R}P^n$.