I'm taking an introductory course in topology and we have a homework exercise to compute the fundamental group of X = $\mathbb{R} \big / \{-1, 1\}$ under the group action $\mathbb{Z}_2\times\mathbb{R} \rightarrow \mathbb{R}$ taking $(n, r)$ to $nr$.
The way I want to approach this is using the Van Kampen theorem(not explicitly stated). Is it correct to think of the equivalence classes of the quotient space as the set, $U$, of positive real numbers and the set, $V$, of negative real numbers (since we multiply by either 1 or -1)?
If so, since the union of $U$ and $V$ is X and their intersection is just $\{ 0 \}$, can we apply the Van Kampen theorem to said sets to conclude that $\pi_1(X) = \pi_1(U)*\pi_1(V) = 0$ in the sense that both $U$ and $V$ are both contractible spaces(or simply connected) and have trivial fundamental group.
An alternative approach would be to use the covering space theory, but I don't think the action is properly discontinuous since $\{0\} \cap n*\{0\} \neq \emptyset$, or any interval containing 0 for that matter. So I don't think that is a better approach. Any hint or clarification to how to use the Van Kampen theorem in this example would be appreciated!
Van Kampen doesn't seem like the way to go here to be honest. First of all, their intersection is not just $0$ under the equivalence relation, they are the same set, so this doesn't really work.
I think a better way to approach this is to find a space homeomorphic to the quotient space. Try $\Bbb R_{\geq 0}$.