nLab non-Hausdorff manifold

Idea

It may superficially seem that every locally Euclidean space is a Hausdorff topological space, since Euclidean space is. While the analogous conclusion is in fact true for genuine “local” properties of Euclidean space (such as the $T_1$-separation axiom, sobriety and local compactness, see this prop.), it is actually wrong for the “non-local” $T_2$= Hausdorff-conditon: There are locally Euclidean spaces which are not Hausdorff topological spaces, such as the line with two origins.

Since a (paracompact/sigma-compact) Hausdorff locally Euclidean space is called a topological manifold, (paracompact/sigma-compact) non-Hausdorff locally Euclidean spaces are sometimes referred to as “non-Hausdorff manifolds” (an instance of the red herring principle).

Examples

The usual example is the line with two origins: the real line with the point $0$ ‘doubled’. Explicitly, this is a quotient space of $\mathbb{R} \times \{a,b\}$ (given the product topology and with $\{a,b\}$ given the discrete topology) by the equivalence relation generated by identifying $(x,a)$ with $(y,b)$ iff $x = y \ne 0$.

References

• 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)