We touched upon the spaces $\mathbb RP^n$ in our topology class. We were asked to think on whether $\mathbb RP^n$ was Hausdorff.
After a bit of contemplation, I came to conclude that the answer is affirmative if I can show that each pair of distinct lines in $\mathbb R^n$ passing through origin can be separated by "cones" containing them. I reckoned that I just need to show the following:
If $x, y\in\mathbb R^n$ such that $\lVert x\rVert, \lVert y\rVert = 1$ and if $\varepsilon > 0$ is such that $B_\varepsilon(x)$ is disjoint with each of $B_\varepsilon(y)$ and $B_\varepsilon(-y)$, then no two points in $B_\varepsilon(x)$ and $B_\varepsilon(y)$ can be scalar multiples of each other.
The answer seems to be yes because of the intuition for $n = 1, 2$ cases. But I am unable to prove this rigorously. Any hints?
