 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.