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geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
and
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, 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 ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
. In
general topology In mathematics, general topology (or point set 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 differ ...
, 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, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, 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 \ and \R \times \ (with a \neq b), obtained by identifying points (x,a) and (x,b) whenever x \neq 0. An equivalent description of the space is to take the
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
\R and replace the origin 0 with two origins 0_a and 0_b. The subspace \R\setminus\ retains its usual Euclidean topology. And a local base of open neighborhoods at each origin 0_i is formed by the sets (U\setminus\)\cup\ with U an open neighborhood of 0 in \R. For each origin 0_i the subspace obtained from \R by replacing 0 with 0_i is an open neighborhood of 0_i homeomorphic to \R. 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 0_a intersects every neighbourhood of 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 In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Conne ...
but not
arc connected Arc may refer to: Mathematics * Arc (geometry), a segment of a differentiable curve ** Circular arc, a segment of a circle * Arc (topology), a segment of a path * Arc length, the distance between two points along a section of a curve * Arc (pr ...
. In particular, to get a path from one origin to the other one can first move left from 0_a to -1 within the line through the first origin, and then move back to the right from -1 to 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 set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...
s need not be compact. For example, the sets 1,0)\cup\ and [-1,0)\cup\ are compact, but their intersection [-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.


Line with many origins

The line with many origins is similar to the line with two origins, but with an arbitrary number of origins. It is constructed by taking an arbitrary set S with the discrete topology and taking the quotient space of \R\times S that identifies points (x,\alpha) and (x,\beta) whenever x\ne 0. Equivalently, it can be obtained from \R by replacing the origin 0 with many origins 0_\alpha, one for each \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 A= 1,0)\cup\\cup(0,1/math> is the set A\cup\ obtained by adding all the origins to A, and that closure is not compact. From being locally Euclidean, such a space is
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
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 \R \times \ \quad \text \quad \R \times \ 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. A simpler example is equ ...
(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, 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 ne ...
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, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, 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, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
in general) and locally Hausdorff (but not Hausdorff in general).


See also

* * *


Notes


References

* * * * {{Topology General topology Manifolds Topology