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

R × {a} and R × {b}

with the equivalence relation

${\displaystyle (x,a)\sim (x,b){\text{ if }}x\neq 0.}$

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 ${\displaystyle 0_{a}}$ in this space can be thought to consist of sets of the form ${\displaystyle \{r\in \mathbb {R} \setminus \{0\}\vert -\varepsilon <r<\varepsilon \}\cup \{0_{a}\}}$, where ${\displaystyle \varepsilon }$ is any positive real number. A similar description of a local base of open neighborhoods of ${\displaystyle 0_{b}}$ is possible. Thus, in this space all neighbourhoods of 0_a intersect all neighbourhoods of 0_b, 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.[1]

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

R × {a} and R × {b}

with the equivalence relation

${\displaystyle (x,a)\sim (x,b){\text{ if }}x<0.}$

This space has a single point for each negative real number r and two points ${\displaystyle x_{a},x_{b}}$ 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.)[2]

Notes

1. Gabard, pp. 4–5
2. Warner, Frank W. (1983). Foundations of Differentiable Manifolds and Lie Groups. New York: Springer-Verlag. p. 164. ISBN 978-0-387-90894-6.

References

• Baillif, Mathieu; Gabard, Alexandre, Manifolds: Hausdorffness versus homogeneity, arXiv:math.GN/0609098v1
• Gabard, Alexandre, A separable manifold failing to have the homotopy type of a CW-complex, arXiv:math.GT/0609665v1