TITLE: An intuitive idea about fundamental group of $\mathbb{RP}^2$
QUESTION [35 upvotes]: 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 $a$, $[2a] = [1]$"

REPLY [5 votes]: Here is a bit more algebraic perspective: if $X$ and $Y$ are path connected and $p: Y\rightarrow X$ is a covering map, then the number of sheets is equal to index of the subgroup $p_*(\pi_1(Y))$ in $\pi_1(X)$. Since $S^2 \rightarrow \mathbb{R}P^2$ is a 2-sheeted universal cover, it follows that $\pi_1(\mathbb{R}P^2)$ has 2 elements (trivial subgroup has index 2).