A manifold is usually by default assumed to have an underlying topological space which is a Hausdorff topological space. If that condition is explicitly dropped, one accordingly speaks of a non-Hausdorff manifold. Compare the red herring principle.
The usual example is the real line with the point ‘doubled’. Explicitly, this is a quotient space of (given the product topology and with given the discrete topology) by the equivalence relation generated by identifying with iff .
Mathieu Baillif, Alexandre Gabard, Manifolds: Hausdorffness versus homogeneity (arXiv:0609098)
Steven L. Kent, Roy A. Mimna, and Jamal K. Tartir, A Note on Topological Properties of Non-Hausdorff Manifolds, (web)