Consider $S^1=\left\{x\in\mathbb{R}^2: \lVert x\rVert=1\right\}$. Now let $P^1$ be obtained from $S^1$ by identifying antipodal points.
I have the following questions:
1) How can I imagine $P^1$? I do not have an idea how I can imagine in in order to get an impression how it looks like.
2) What could be meant by $[x,-x]\in P^1$?
3) An element $x\in S^1$ can be written as $x=e^{i\phi}$. Is there a similar way to write an element of of $P^1$ in polar coordinates?
You can image it as a $S^{1}$.
Note that if you have $S^{1}$ and identify all points except those intersecting the $y$ axis you get a half circle. Then you need to identify the points that intersect the $y$ axis closing the circle again.
The notation refers to the fact that your points are not points in $\mathbb{R}^{2}$ but equivalence classes of points in $S^{1}$ identify by antipodal points.
Yes, because as I said the new space is basically $S^{1}$.
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