In geometry and topology, it is a usual axiom of a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces

locally homeomorphic In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If f : X \to Y is a local homeomorphism, X is said to be an � ...

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 Quotient space may refer to a quotient set when the sets under consideration are considered as spaces. In particular: *Quotient space (topology), in case of topological spaces * Quotient space (linear algebra), in case of vector spaces *Quotient ...

of two copies of the real line

\R \times \ \quad \text \quad \R \times \

with the

equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...

(x, a) \sim (x, b) \quad \text \; x \neq 0.

This space has a single point for each nonzero real number

r

and two points

0_a

and

0_b.

A local base of open neighborhoods of

0_a

in this space can be thought to consist of sets of the form

\ \cup \,

where

\varepsilon > 0

is any positive real number. A similar description of a local base of open neighborhoods of

0_b

is possible. Thus, in this space all neighbourhoods of

0_a

intersect all neighbourhoods of

0_b,

so the space is non-Hausdorff. Further, the line with two origins does not have the homotopy type of a CW-complex, or of any Hausdorff space.Gabard, pp. 4–5

Branching line

Similar to the line with two origins is the branching line. This is the

of two copies of the real line

\R \times \ \quad \text \quad \R \times \

with the

(x, a) \sim (x, b) \quad \text \; x < 0.

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 In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...

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

to Euclidean space, they are locally metrizable (but not metrizable) and locally Hausdorff (but not Hausdorff).

Notes

References

* * {{Topology General topology Manifolds Separation axioms Topology