Difference between the sphere and RP^2 in the generators and relations of $\pi_1$

Question

Here are two drawings we had that I did not quite well understand why they are correct:

Here are my questions:

1-I think the drawing of $RP^2$ is meant to be a circle not a disk, am I correct?

2- what should be the generators of $S^2$? And how this leads us to conclude that $\pi_1$ of it is zero?

Could someone clarify these points for me please?

Answer 1

• For $RP^2$, its classic definition is indeed the disc quotiented by the relation on the 'edges', so here it indeed is disk, and not a circle
• $S^2$ is simply connected, which precisely means that $\pi_1(S^2)=\{id\}$ is trivial. We can see this either via more general results, or by showing a path on $S^2$ is going to cover some point $x$ finitely many times, then moving the path away from $x$. Then the path has image in $S^2\backslash \{x\}$ which is homeomorphic to $\mathbb{R}^2$ which is simply connected so the path is homotopic to the constant path.