# What are the sufficient conditions for quotient of Hausdorff space to again be a Hausdorff space?

Initial question in the paper was Is the quotient of Hausdorff space always Hausdorff?.

My attempt so far: I have been able to find a counter example:

Let $$\Bbb Z$$/2 act on $$\Bbb R$$×{0} ∪ $$\Bbb R$$×{1} by g ⋅ (t,0) = (t,1) and g ⋅ (t,1) = (t,0) if t ≠ 0 and g ⋅ (0,0) = (0,0) and g ⋅ (1,1) = 1. Then the quotient space is the line with two origins which is not Hausdorff.

So I understand that initial question is false. I am not particularly happy about the group theoretic nature of the counter example. Which leads me to ask what are the sufficient conditions such that quotient of Hausdorff would be Hausdorff.

Let $$X$$ be a hausdorff space and let $$\sim$$ be an equivalence relation on $$X$$. Suppose that the quotient map $$\eta \colon X \rightarrow X/{\sim}$$ is open. Then $$X/{\sim}$$ is hausdorff iff $$R \subset X \times X$$ is closed, where $$R$$ is the set that defines $$\sim$$.