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.
The most familiar non-Hausdorff manifold is the line with two origins,[1] or bug-eyed line.
This is the quotient space of two copies of the real line,
), obtained by identifying points
An equivalent description of the space is to take the real line
retains its usual Euclidean topology.
And a local base of open neighborhoods at each origin
[1] 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
intersects every neighbourhood of
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.
[2] The line with many origins[3] is similar to the line with two origins, but with an arbitrary number of origins.
It is constructed by taking an arbitrary set
with the discrete topology and taking the quotient space of
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
obtained by adding all the origins to
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.
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
for every non-negative number: it has a "fork" at zero.
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.
)[4] 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).