Let $I=[0,1],X=I/\{0,1\}$.Prove that $\pi_1(X,\bar{0})$ is a free abelian group with one generator.
I note that $X$ is homeomorphic to $S^1$ given by $h(x)=(\cos(2\pi x),\sin(2\pi x))$,then the fundamental group of $X$ and $S^1$ based at $\bar{0}$ is equal and isomorphic to $\mathbb{Z}$. Then, I'm wondering what is exactly the one generator that generates the free abelian group?
$\Bbb Z$ has $1$ as a generator; the other generator is $-1$.