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I am trying to prove that a quotient space is Hausdorff but I don't have a lot of technique with this type of exercises. Can someone provide me a hint to start?

Let $\tau$ be a complex number with $\operatorname{Im}\tau>0$, $\Lambda=\{n+m\tau:n,m\in\mathbb{Z}\}$ and define $E=\mathbb{C}/\Lambda$. Let $\pi:\mathbb{C}\to E$ be the quotient map: $\pi(z)=z+\Lambda$ and equip $E$ with the quotient topology: $W\subseteq E$ is open if $\pi^{-1}(W)$ is open in $\mathbb{C}$. This makes $E$ into a topological space. Prove that $E$ is Hausdorff.

I fixed $z_1+\Lambda$, $z_2+\Lambda\in E$ distinct and wrote them as $\pi(z_1)$ and $\pi(z_2)$, respectively. Now what should I do?

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Hints:

So you have a lattice $\;\Lambda\;$ and a torus $\;\Bbb C/\Lambda\;$ (or elliptic curve over the complex field). Take for example the diagonal

$$\Delta:=\left\{\;(x,x)\;/\;x\in E\;\right\}$$

You must be sure you understand $\;E\;$ is Hausdorff iff $\;\Delta\;$ is closed in the product topology. But it is now not hard to show any converging sequence in $\;\Delta\;$ converges to a sequence in $\;\Delta\;$, too.

Or also: let $\;(a,b)\in E^2\setminus\Delta\;$ , so $\;a\neq b\pmod{\Delta}\;$ . Take nice representative of the respective cosets and use the fact that $\;\Bbb C\;$ is Hausdorff, and now use the quotient map and etc.

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