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
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
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 many ...
. 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, geomet ...
, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces
locally homeomorphic
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure.
If f : X \to Y is a local homeomorphism, X is said to be an à ...
to
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, 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
with the
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
This space has a single point for each nonzero real number
and two points
and
A local base of open neighborhoods of
in this space can be thought to consist of sets of the form
where
is any positive real number. A similar description of a local base of open neighborhoods of
is possible. Thus, in this space all neighbourhoods of
intersect all neighbourhoods of
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 cla ...
, 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
with the
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
This space has a single point for each negative real number
and two points
for every non-negative number: it has a "fork" at zero.
Etale space
The
etale space
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
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 new ...
property.)
Properties
Because non-Hausdorff manifolds are
locally homeomorphic
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure.
If f : X \to Y is a local homeomorphism, X is said to be an à ...
to
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, 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) ...
) and
locally Hausdorff (but not
Hausdorff).
See also
*
*
*
Notes
References
*
*
{{Topology
General topology
Manifolds
Separation axioms
Topology