Given the following exercise:
Consider on $\mathbb{R}^2$ the subsets:
Now the exercise asks one to give a $C^{\infty}$ atlas on S.
To define open sets on a quotient space obtained from an equivalence relation you take the canonical projection $\pi : E \rightarrow $E/~ and consider a set $V \subset $ E/~ open if and only if $\pi^{-1}(V)$ is open in E.
In the solution, one chart that he constructs uses a coordinate domain of the form:
$U = \{[(x,0)] : x \lt 0 \} \cup \{[(x,1)] : x \geq 0 \}$
but I don't see how $\{[(x,1)] : x \geq 0 \}$ is open in the subspace E, since there is no set $U \subset \mathbb{R}^2$ such that $E_2 \cap U = \{(x,1) : x \geq 0\}$, so it can't be open in E/~.
Maybe I completely misunderstand something about quotient spaces, could somebody clarify this for me please?