# Open sets in quotient space defined by equivalence relation

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?

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