A non-Hausdorff manifold is a locally Euclidean topological space for which distinct points need not have disjoint neighborhoods. The real line with two origins is a basic example. Nonseparated algebraic schemes exhibit analogous gluing phenomena, but are not generally topological manifolds.
Take with its Euclidean topology and identify with for every , 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 and , so a nonzero 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.
