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Given the following exercise:

Consider on $\mathbb{R}^2$ the subsets:

enter image description here

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?

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    $\begingroup$ No one is claiming that $\{[(x,1)]\;:\:x\geq0\}$ is open. Its union with $\{[(x,0)]\;:\;x<0\}, $however, is. $\endgroup$ – Amitai Yuval Nov 3 '18 at 13:38
  • $\begingroup$ I see, thank you. $\endgroup$ – eager2learn Nov 3 '18 at 13:58

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