non-Hausdorff manifold
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In
geometry and topology In mathematics, geometry and topology is an umbrella term for the historically distinct disciplines of geometry and topology, as general frameworks allow both disciplines to be manipulated uniformly, most visibly in local to global theorems in Ri ...
, it is a usual axiom of a manifold to be a
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
. In
general topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometri ...
, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, 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 \times \ \quad \text \quad \R \times \ with the equivalence relation (x, a) \sim (x, b) \quad \text \; 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 \ \cup \, 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 the space is non-Hausdorff. Further, the line with two origins does not have the homotopy type of a
CW-complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This clas ...
, or of any Hausdorff space.Gabard, pp. 4–5


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 \times \ \quad \text \quad \R \times \ with the equivalence relation (x, a) \sim (x, b) \quad \text \; 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 Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * ''The Sheaf'', a student-run newspaper se ...
, 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 In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a n ...
property.)


Properties

Because non-Hausdorff manifolds are locally homeomorphic to
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, they are locally metrizable (but not
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
) and locally Hausdorff (but not Hausdorff).


See also

* * *


Notes


References

* * {{Topology General topology Manifolds Separation axioms Topology