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 $a$, $[2a] = [1]$"

Try watching Your palm is a spinor on youtube. This move is part of a traditional phillipine dance – watch about 40 seconds into the clip.
As you go from the performer's more or less stationary shoulder to the hand that holds the glass, you are in fact following a homotopy from the trivial loop to the loop that rotates 720 degrees around a vertical axis.
The move is not hard to learn. But try it with an empty glass at first.
Edit: I neglected to add that this is really about $\mathrm{SO}(3)\simeq\mathbb{R}P^3$, not $\mathbb{R}P^2$. It's the same sort of thing going on, really. To see that $\mathrm{SO}(3)\simeq\mathbb{R}P^3$, think of a rotation as specified by a vector $x\in\mathbb{R}$ with $\|x\|\le\pi$, the direction giving the axis and the length the angle of rotation in the positive direction, as seen from the positive end of the axis. This identifies antipodal points on the sphere of radius $\pi$, thus turning the closed ball into a projective 3-space.
