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

\mathbb {R} \times \{a\}\quad {\text{ and }}\quad \mathbb {R} \times \{b\}

with the equivalence relation

(x,a)\sim (x,b)\quad {\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 $0_{a}$ in this space can be thought to consist of sets of the form

\{r\in \mathbb {R} \setminus \{0\}~:~-\varepsilon <r<\varepsilon \}\cup \{0_{a}\},

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 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

\mathbb {R} \times \{a\}\quad {\text{ and }}\quad \mathbb {R} \times \{b\}

with the equivalence relation

(x,a)\sim (x,b)\quad {\text{ if }}\;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 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

^ Gabard, pp. 4–5
^ 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 (2006), Manifolds: Hausdorffness versus homogeneity, arXiv:math.GN/0609098v1, Bibcode:2006math......9098B
Gabard, Alexandre (2006), A separable manifold failing to have the homotopy type of a CW-complex, arXiv:math.GT/0609665v1, Bibcode:2006math......9665G

[1] Gabard, pp. 4–5

[Warner-2] Warner, Frank W. (1983). Foundations of Differentiable Manifolds and Lie Groups. New York: Springer-Verlag. p. 164. ISBN 978-0-387-90894-6.

[1]

[2]