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, and (with a \neq b), obtained by identifying points (x,a) and (x,b) whenever x \neq 0. An equivalent description of the space is to take the real line \R and replace the origin 0 with two origins 0_a and 0_b. The subspace retains its usual Euclidean topology. And a local base of open neighborhoods at each origin 0_i is formed by the sets with U an open neighborhood of 0 in \R. For each origin 0_i the subspace obtained from \R by replacing 0 with 0_i is an open neighborhood of 0_i homeomorphic to \R. Since every point has a neighborhood homeomorphic to the Euclidean line, the space is locally Euclidean. In particular, it is locally Hausdorff, in the sense that each point has a Hausdorff neighborhood. But the space is not Hausdorff, as every neighborhood of 0_a intersects every neighbourhood of 0_b. It is however a T1 space. The space is second countable. The space exhibits several phenomena that do not happen in Hausdorff spaces: The space does not have the homotopy type of a CW-complex, or of any Hausdorff space.

Line with many origins

The line with many origins is similar to the line with two origins, but with an arbitrary number of origins. It is constructed by taking an arbitrary set S with the discrete topology and taking the quotient space of \R\times S that identifies points (x,\alpha) and (x,\beta) whenever x\ne 0. Equivalently, it can be obtained from \R by replacing the origin 0 with many origins 0_\alpha, one for each The neighborhoods of each origin are described as in the two origin case. If there are infinitely many origins, the space illustrates that the closure of a compact set need not be compact in general. For example, the closure of the compact set is the set obtained by adding all the origins to A, and that closure is not compact. From being locally Euclidean, such a space is locally compact in the sense that every point has a local base of compact neighborhoods. But the origin points do not have any closed compact neighborhood.

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 x_a, x_b 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. (The etale space is Hausdorff if it is a sheaf of functions with some sort of analytic continuation property.)

Properties

Because non-Hausdorff manifolds are locally homeomorphic to Euclidean space, they are locally metrizable (but not metrizable in general) and locally Hausdorff (but not Hausdorff in general).