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, ^[1] or bug-eyed line. This is the quotient space of two copies of the real line, ${\displaystyle \mathbb {R} \times \{a\}}$ and ${\displaystyle \mathbb {R} \times \{b\}}$ (with ${\displaystyle a\neq b}$), obtained by identifying points ${\displaystyle (x,a)}$ and ${\displaystyle (x,b)}$ whenever ${\displaystyle x\neq 0.}$

An equivalent description of the space is to take the real line ${\displaystyle \mathbb {R} }$ and replace the origin ${\displaystyle 0}$ with two origins ${\displaystyle 0_{a}}$ and ${\displaystyle 0_{b}.}$ The subspace ${\displaystyle \mathbb {R} \setminus \{0\}}$ retains its usual Euclidean topology. And a local base of open neighborhoods at each origin ${\displaystyle 0_{i}}$ is formed by the sets ${\displaystyle (U\setminus \{0\})\cup \{0_{i}\}}$ with ${\displaystyle U}$ an open neighborhood of ${\displaystyle 0}$ in ${\displaystyle \mathbb {R} .}$

For each origin ${\displaystyle 0_{i}}$ the subspace obtained from ${\displaystyle \mathbb {R} }$ by replacing ${\displaystyle 0}$ with ${\displaystyle 0_{i}}$ is an open neighborhood of ${\displaystyle 0_{i}}$ homeomorphic to ${\displaystyle \mathbb {R} .}$ ^[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 ${\displaystyle 0_{a}}$ intersect every neighbourhood of ${\displaystyle 0_{b}.}$ It is however a T₁ space.

The space is second countable.

The space exhibits several phenomena that do not happen in Hausdorff spaces:

• The space is path connected but not arc connected. In particular, to get a path from one origin to the other one can first move left from ${\displaystyle 0_{a}}$ to ${\displaystyle -1}$ within the line through the first origin, and then move back to the right from ${\displaystyle -1}$ to ${\displaystyle 0_{b}}$ within the line through the second origin. But it is impossible to join the two origins with an arc, which is an injective path; intuitively, if one moves first to the left, one has to eventually backtrack and move back to the right.

• The intersection of two compact sets need not be compact. For example, the sets ${\displaystyle [-1,0)\cup \{0_{a}\}}$ and ${\displaystyle [-1,0)\cup \{0_{b}\}}$ are compact, but their intersection ${\displaystyle [-1,0)}$ is not.

• The space is locally compact in the sense that every point has a local base of compact neighborhoods. But the line through one origin does not contain a closed neighborhood of that origin, as any neighborhood of one origin contains the other origin in its closure. So the space is not a regular space, and even though every point has at least one closed compact neighborhood, the origin points do not admit a local base of closed compact neighborhoods.

The space does not have the homotopy type of a CW-complex, or of any Hausdorff space. ^[2]

Line with many origins

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 ${\displaystyle S}$ with the discrete topology and taking the quotient space of ${\displaystyle \mathbb {R} \times S}$ that identifies points ${\displaystyle (x,\alpha )}$ and ${\displaystyle (x,\beta )}$ whenever ${\displaystyle x\neq 0.}$ Equivalently, it can be obtained from ${\displaystyle \mathbb {R} }$ by replacing the origin ${\displaystyle 0}$ with many origins ${\displaystyle 0_{\alpha },}$ one for each ${\displaystyle \alpha \in S.}$ The neighborhoods of each origin are described as in the two origin case.

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 ${\displaystyle A=[-1,0)\cup \{0_{\alpha }\}\cup (0,1]}$ is the set ${\displaystyle A\cup \{0_{\beta }:\beta \in S\}}$ obtained by adding all the origins to ${\displaystyle A}$, and that closure is not compact. 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.

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

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

with the equivalence relation

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

This space has a single point for each negative real number ${\displaystyle r}$ 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.) ^[4]

Properties

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).

See also

• List of topologies  – List of concrete topologies and topological spaces
• Locally Hausdorff space
• Separation axiom  – Axioms in topology defining notions of "separation"

Notes

1. Munkres 2000, p. 227.
2. Gabard 2006, Proposition 5.1.
3. Lee 2011, Problem 4-22, p. 125.
4. 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 (2008). "Manifolds: Hausdorffness versus homogeneity". Proceedings of the American Mathematical Society. 136 (3): 1105–1111. arXiv: math/0609098 . doi: 10.1090/S0002-9939-07-09100-9 .
• Gabard, Alexandre (2006), A separable manifold failing to have the homotopy type of a CW-complex, arXiv: math.GT/0609665v1 , Bibcode:2006math......9665G
• Lee, John M. (2011). Introduction to topological manifolds (Second ed.). Springer. ISBN   978-1-4419-7939-1.
• Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN   978-0-13-181629-9. OCLC   42683260.