Corollary $\bf7.15.$ The real projective space $\mathbb{R}P^n$ is second countable.
How I can prove this corollary 7.15.I consider that ı can apply the corollary 7.10.But I could not can you help me.
Corollary $\bf7.10.$ If $\,\sim\,$ is an open equivalence relation on a second-countable space $S$, then the quotient space $S/\sim$ is second countable.
Recall that $S^n\subset\mathbb{R}^{n+1}$ is second countable as it is a subspace of a second countable space. Can you see how to use your corollary 7.10, along with this fact to show that $\mathbb{R}P^n$ is second countable?