homeomorphic topology of quotient space of $S^1$

Question

I got stuck on the problem about quotient space from General Topology of Stephen Williard. Here is the problem:

Let $\sim$ be the equivalence relation $x \sim y$ iff $x$ and $y$ are diametrically opposite, on $S^1$. Which topology is the quotient space $S^1/\sim$ homeomorphic to?

I tried to build a continuous function $S^1$ such that 2 points which are diametrically have the same images, but I couldn't find it. For each point in $S^1$, we can write it as $(\cos(\phi), \sin(\phi))$, then what is the function which satisfies the previous requirement. Can anyone help me with this? I really appreciate.

Answer 1

Take a rubber band; that’s your $S^1$. Now fold it into a figure eight: 8. Finally, fold the $8$ about its horizontal midline to get a double circle. Thus, you’ve gone from O to 8 to doubled o. Check that this brings diametrically opposite points of the original O together: they’re simply on different copies of the o.

Answer 2

The space you will get is $\mathbb{RP}^1$, the one dimensional real projective space which is by definition of all lines through the origin in $\mathbb{R}^2$.