# 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.

## 1 Answer

Let me give you a condition that occurs quite naturally and does not deal with groups:

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 $$\sim$$.