View source for 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''',{{sfn|Munkres|2000|p=227}} or '''bug-eyed line'''.  This is the [[quotient space (topology)|quotient space]] of two copies of the real line,
<math>\R \times \{a\}</math> and <math>\R \times \{b\}</math> (with <math>a \neq b</math>), obtained by identifying points <math>(x,a)</math> and <math>(x,b)</math> whenever <math>x \neq 0.</math>

An equivalent description of the space is to take the [[real line]] <math>\R</math> and replace the origin <math>0</math> with two origins <math>0_a</math> and <math>0_b.</math>  The subspace <math>\R\setminus\{0\}</math> retains its usual Euclidean topology.  And a [[local base]] of open neighborhoods at each origin <math>0_i</math> is formed by the sets <math>(U\setminus\{0\})\cup\{0_i\}</math> with <math>U</math> an open neighborhood of <math>0</math> in <math>\R.</math>

For each origin <math>0_i</math> the subspace obtained from <math>\R</math> by replacing <math>0</math> with <math>0_i</math> is an open neighborhood of <math>0_i</math> homeomorphic to <math>\R.</math>{{sfn|Munkres|2000|p=227}}  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 <math>0_a</math> intersects every neighbourhood of <math>0_b.</math>  It is however a [[T1 space|T<sub>1</sub> 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 <math>0_a</math> to <math>-1</math> within the line through the first origin, and then move back to the right from <math>-1</math> to <math>0_b</math> 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 set]]s need not be compact.  For example, the sets <math>[-1,0)\cup\{0_a\}</math> and <math>[-1,0)\cup\{0_b\}</math> are compact, but their intersection <math>[-1,0)</math> 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.{{sfn|Gabard|2006|loc=Proposition 5.1}}

===Line with many origins===

The '''line with many origins'''{{sfn|Lee|2011|loc=Problem 4-22, p. 125}} is similar to the line with two origins, but with an arbitrary number of origins.  It is constructed by taking an arbitrary set <math>S</math> with the discrete topology and taking the quotient space of <math>\R\times S</math> that identifies points <math>(x,\alpha)</math> and <math>(x,\beta)</math> whenever <math>x\ne 0.</math>  Equivalently, it can be obtained from <math>\R</math> by replacing the origin <math>0</math> with many origins <math>0_\alpha,</math> one for each <math>\alpha\in S.</math>  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 <math>A=[-1,0)\cup\{0_\alpha\}\cup(0,1]</math> is the set <math>A\cup\{0_\beta:\beta\in S\}</math> obtained by adding all the origins to <math>A</math>, 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.

{{anchor|Branching line}}
===Branching line===

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

This is the [[Quotient space (topology)|quotient space]] of two copies of the real line
<math display=block>\R \times \{a\} \quad \text{ and } \quad \R \times \{b\}</math>
with the [[equivalence relation]]
<math display=block>(x, a) \sim (x, b) \quad \text{ if } \; x < 0.</math>

This space has a single point for each negative real number <math>r</math> and two points <math>x_a, x_b</math> for every non-negative number: it has a "fork" at zero.

===Etale space===

The [[etale space]] of a [[sheaf (mathematics)|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.)<ref name="Warner">{{cite book |last1=Warner |first1=Frank W. |title=Foundations of Differentiable Manifolds and Lie Groups |url=https://archive.org/details/foundationsdiffe00warn |url-access=limited |date=1983 |publisher=Springer-Verlag |location=New York |isbn=978-0-387-90894-6 |page=[https://archive.org/details/foundationsdiffe00warn/page/n171 164]}}</ref>

==Properties==

Because non-Hausdorff manifolds are [[locally homeomorphic]] to [[Euclidean space]], they are [[Locally metrizable space|locally metrizable]] (but not [[Metrizable space|metrizable]] in general) and [[Locally Hausdorff space|locally Hausdorff]] (but not [[Hausdorff space|Hausdorff]] in general).

==See also==

* {{annotated link|List of topologies}}
* {{annotated link|Locally Hausdorff space}}
* {{annotated link|Separation axiom}}

==Notes==

{{reflist}}

==References==

* {{cite journal |last1=Baillif |first1=Mathieu |last2=Gabard |first2=Alexandre |title=Manifolds: Hausdorffness versus homogeneity |journal=Proceedings of the American Mathematical Society |date=2008 |volume=136 |issue=3 |pages=1105–1111 |doi=10.1090/S0002-9939-07-09100-9|doi-access=free |arxiv=math/0609098 }}
* {{citation
|last=Gabard|first=Alexandre
|title=A separable manifold failing to have the homotopy type of a CW-complex
|year=2006
|arxiv=math.GT/0609665v1
|bibcode=2006math......9665G}}
* {{cite book |last1=Lee |first1=John M. |title=Introduction to topological manifolds |date=2011 |publisher=Springer |isbn=978-1-4419-7939-1 |edition=Second}}
* {{Munkres Topology|edition=2}} <!--{{sfn|Munkres|2000|p=}}-->

{{Manifolds}}
{{Topology}}

[[Category:General topology]]
[[Category:Manifolds]]
[[Category:Topology]]