Imagining $S^1$ with antipodal points identified

Question

How can I imagine $S^1$ with antipodal points identified? Are there more ways, say a few rendering nearby points nearby, and some other not?

And yet a smooth version?

Answer 1

Big Hint: Denote elements of $S^1/\{\pm 1\}$ by $[z]$ where $z$ is a representative in $S^1$. Then, consider the map $f:S^1/\{\pm 1\}\to S^1$, $[z]\mapsto z^2$. Show that this is a diffeomorphism.