An intuitive idea about fundamental group of $\mathbb{RP}^2$ Someone can explain me with an example, what is the meaning of $\pi(\mathbb{RP}^2,x_0) \cong \mathbb{Z}_2$? 
We consider the real projective plane as a quotient of the disk.
I didn't receive an exhaustive answer to this question from my teacher, in fact he said that the loop $2a$ with base point $P$ is homotopically equivalent to the "constant loop" with base point $P$. but this doesn't solve my doubts.
Obviously I can calculate it, so the problem is NOT how to calculate it using Van Kampen theorem, but I need to get an idea of "why for every loop $a$, $[2a] = [1]$"

You can see another set of related pictures here, which gives the script for this video 
Pivoted Lines and the Mobius Band (1.47MB). 
The term "Pivoted Lines" is intended to be a non technical reference to the fact that we are discussing rotations, and their representations. The video shows the "identification" of the Projective Plane as a Mobius Band and a disk, the identification being shown by a  point moving from one to the other. Then the point makes a loop twice round the Mobius Band, 
                 
as  in the above, and this loop moves off the Band onto the disk and so to a point. Thus we are representing motion of motions!
