Line with two origins

Take $\mathbb{R}\times \{0,1\}$ with its Euclidean topology and identify $(t,0)$ with $(t,1)$ for every $t\ne 0$, retaining two distinct origins. This quotient space is locally homeomorphic to the real line but is not Hausdorff. Any open neighbourhoods of its origins have preimages containing $(-\epsilon ,\epsilon )\times \{0\}$ and $(-\delta ,\delta )\times \{1\}$, so a nonzero $|t|<\min (\epsilon ,\delta )$ gives a shared image point. All equivalence classes in the original metric space are finite and closed. This shows that closed equivalence classes do not suffice for a Hausdorff quotient space.