A simple question about an homeomorphism.

Question

I already proved that the real projective line $\mathbb{R}P^1$ is homeomorphic to the quotient $S^1/\sim$, where $\sim$ is an equivalence relation in $S^1$ which identifies antipodal points.

Question. How can prove that $S^1/ \sim$ is homeomorphic to the closed upper semicircle with the two endpoints identified?

Thanks!

Answer 1

Let $p : S^1 \to X = S^1/\sim$ be the quotient map and $q : S^1_+ \to X$ be the restriction to the closed upper semicircle. It is a closed map since $S^1_+$ is compact and $X$ is Hausdorff, thus a quotient map. Clearly it identifies precisely the endpoints of $S^1_+$.