= Line with two origins
Take $\mathbb R\times\{0,1\}$ with its <Euclidean topology> and identify $(t,0)$ with $(t,1)$ for every $t\neq0$, 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 $(-\varepsilon,\varepsilon)\times\{0\}$ and $(-\delta,\delta)\times\{1\}$, so a nonzero $|t|<\min(\varepsilon,\delta)$ 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>.
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