A question about the real projective space.

Question

I came across the definition of the $n$-dimensional real projective space $\Bbb RP^n$ in the lecture notes given by our instructor. The definition is as follows $:$

Let us take the $n$-sphere $\Bbb S^n$ in $\Bbb R^{n+1}$ and let it be acted by $C_2 = \{1,\tau\},$ the cyclic group of order $2$ by $$\tau \cdot x = -x,\ x \in \Bbb S^n.$$ Then $\Bbb R P^n : = \Bbb S^n/C_2,$ the space where we identify antipodal points (diametrically opposite points) of $\Bbb S^n.$ So $\Bbb R P^n$ is essentially lines in $\Bbb R^{n+1}$ passing through the origin.

This is where I get stuck. How do I think of the space $\Bbb S^n$ with the antipodal points being identified as being equivalent to the space of all lines in $\Bbb R^{n+1}$ passing through the origin? Can anybody give me some geometric intuition of visualizing that space? Any help in this regard will be highly solicited.

Thanks in advance.

Answer 1

In fact, there is a bijection between the set of $\mathbb R^{n+1}$ lines passing through the origin and $\Bbb R P^n$:

A line passing through the origin intersects $\mathbb S^n$ at exactly two antipodal points, i.e. one point of $\Bbb R P^n$. And given a point in $\Bbb R P^n$, you can get two antipodal points of $\mathbb S^n$ which define a single line passing through the origin.

This is what is meant with the word identification.

Answer 2

For each line $L \subset \mathbb R^{n+1}$ through the origin, the intersection $L \cap \mathbb S^n$ is an antipodal pair of points.

Conversely, for each antipodal pair of points $\{x,-x\} \subset \mathbb S^n$, the line in $\mathbb R^{n+1}$ that passes through the two points $x$ and $-x$ also passes through the origin.

Therefore, we obtain a bijection between the set of lines in $\mathbb R^{n+1}$ that pass through the origin and the set of antipodal pairs of points in $\mathbb S^n$. The "equivalence" referred to in your post is simply a reference to this bijection.