TITLE: Quotient space of S1 is homeomorphic to S1
QUESTION [5 upvotes]: $S^1=\{z\in\mathbb{C}\mid |z|=1\}$, let $w\sim z$ iff $w=z\vee w=-z$ (identifying antipodal points). Prove $S^1/\sim$ is homeomorphic to $S^1$. Which function should be used to construct a homeomorphism? I am not good at analysis. Thanks!

REPLY [2 votes]: $f:S^1\to S^1,\ e^{it}\mapsto e^{2it}$, where $t\in[0,2\pi]$, is the function you should use. Note that this function itself is continuous because it is induced by the map $t\mapsto e^{2it}$ which gives the same value for $t=0$ and $t=2\pi$. Once seen that it is well-defined, you can of course write it as $f(z)=z^2$.