Non-Hausdorff manifold


In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to Euclidean space, but not necessarily Hausdorff.

Examples

Line with two origins

The most familiar non-Hausdorff manifold is the line with two origins, or bug-eyed line.
This is the quotient space of two copies of the real line
with the equivalence relation
This space has a single point for each nonzero real number r and two points 0a and 0b. A local base of open neighborhoods of in this space can be thought to consist of sets of the form, where is any positive real number. A similar description of a local base of open neighborhoods of is possible. Thus, in this space all neighbourhoods of 0a intersect all neighbourhoods of 0b, so it is non-Hausdorff.
Further, the line with two origins does not have the homotopy type of a CW-complex, or of any Hausdorff space.

Branching line

Similar to the line with two origins is the branching line.
This is the quotient space of two copies of the real line
with the equivalence relation
This space has a single point for each negative real number r and two points for every non-negative number: it has a "fork" at zero.

Etale space

The etale space of a sheaf, such as the sheaf of continuous real functions over a manifold, is a manifold that is often non-Hausdorff.