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

(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 $0_{a}$ in this space can be thought to consist of sets of the form $\{r\in \mathbb {R} \setminus \{0\}\vert -\varepsilon <r<\varepsilon \}\cup \{0_{a}\}$ , where $\varepsilon$ 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

R × {a} and R × {b}

with the equivalence relation

(x,a)\sim (x,b){\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, 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

[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.

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