Non-Hausdorff quotient of the real line by rational translation
= Non-Hausdorff quotient of the real line by rational translation
Identify $x,y\in\mathbb R$ when $x-y\in\mathbb Q$. Every nonempty saturated open subset of $\mathbb R$ meets every other one because rational translates of any open interval meet any prescribed nonempty open interval. The quotient $\mathbb R/\mathbb Q$ has multiple points but no two nonempty disjoint open subsets, so it is not Hausdorff.