In _{2} space is a _{2}) is the most frequently used and discussed. It implies the uniqueness of

_{1} but is not Hausdorff is the cofinite topology defined on an _{1} spaces in which every convergent sequence has a unique limit.

_{1}, meaning that all singletons are closed. Similarly, preregular spaces are R_{0}. Every Hausdorff space is a

_{0} condition.
These are also the spaces in which completeness makes sense, and Hausdorffness is a natural companion to completeness in these cases.
Specifically, a space is complete if and only if every Cauchy net has at ''least'' one limit, while a space is Hausdorff if and only if every Cauchy net has at ''most'' one limit (since only Cauchy nets can have limits in the first place).

topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

and related branches of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a Hausdorff space, separated space or Ttopological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

where for any two distinct points there exist neighbourhoods of each which are from each other. Of the many separation axiom
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes ...

s that can be imposed on a topological space, the "Hausdorff condition" (Tlimits
Limit or Limits may refer to:
Arts and media
* Limit (music)
In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre (music), genre of music, or the harmonies that can be made using a particular ...

of sequence
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s, net
Net or net may refer to:
Mathematics and physics
* Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, ...

s, and filter
Filter, filtering or filters may refer to:
Science and technology Device
* Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass
** Filter (aquarium), critical ...

s.
Hausdorff spaces are named after Felix Hausdorff
Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...

, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom
An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

.
Definitions

Points $x$ and $y$ in a topological space $X$ can be ''separated by neighbourhoods
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric obj ...

'' if there exists
In predicate logic, an existential quantification is a type of Quantification (logic), quantifier, a logical constant which is interpretation (logic), interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by ...

a neighbourhood
A neighbourhood (British English, Hiberno-English, Hibernian English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographicall ...

$U$ of $x$ and a neighbourhood $V$ of $y$ such that $U$ and $V$ are ($U\backslash cap\; V=\backslash empty$).
$X$ is a Hausdorff space if all distinct points in $X$ are pairwise neighbourhood-separable. This condition is the third separation axiom
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes ...

(after $T\_0,T\_1$), which is why Hausdorff spaces are also called $T\_2$ spaces. The name ''separated space'' is also used.
A related, but weaker, notion is that of a preregular space. $X$ is a preregular space if any two topologically distinguishable
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struct ...

points can be separated by disjoint neighbourhoods. Preregular spaces are also called ''$R\_1$ spaces''.
The relationship between these two conditions is as follows. A topological space is Hausdorff if and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...

it is both preregular (i.e. topologically distinguishable points are separated by neighbourhoods) and Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovie ...

(i.e. distinct points are topologically distinguishable). A topological space is preregular if and only if its Kolmogorov quotient
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...

is Hausdorff.
Equivalences

For a topological space ''$X$'', the following are equivalent: * $X$ is a Hausdorff space. * Limits of nets in ''$X$'' are unique. * Limits offilters
Filter, filtering or filters may refer to:
Science and technology Device
* Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass
** Filter (aquarium), critical ...

on ''$X$'' are unique.
* Any singleton set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

$\backslash \; \backslash subset\; X$ is equal to the intersection of all closed neighbourhoods of ''$X$''. (A closed neighbourhood of ''$X$'' is a closed set
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

that contains an open set containing ''x''.)
* The diagonal ''$\backslash Delta\; =\; \backslash $'' is closed as a subset of the product space
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...

''$X\; \backslash times\; X$''.
* Any injection from the discrete space with two points to ''$X$'' has the lifting property
In mathematics, in particular in category theory, the lifting property is a property of a pair of morphism (category theory), morphisms in a category (mathematics), category. It is used in homotopy theory within algebraic topology to define properti ...

with respect to the map from the finite topological space with two open points and one closed point to a single point.
Examples and non-examples

Almost all spaces encountered inanalysis
Analysis is the process of breaking a complex topic or substance
Substance may refer to:
* Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes
* Chemical substance, a material with a definite chemical composit ...

are Hausdorff; most importantly, the real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s (under the standard metric topology
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

on real numbers) are a Hausdorff space. More generally, all metric space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

s are Hausdorff. In fact, many spaces of use in analysis, such as topological group
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s and topological manifold In topology, a branch of mathematics, a topological manifold is a topological space which locally resembles real numbers, real ''n''-dimension (mathematics), dimensional Euclidean space. Topological manifolds are an important class of topological sp ...

s, have the Hausdorff condition explicitly stated in their definitions.
A simple example of a topology that is Tinfinite set
In set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...

.
Pseudometric space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in the construction of Hausdorff gauge space
In the mathematical
Mathematics (from Greek
Greek may refer to:
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Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population i ...

s. Indeed, when analysts run across a non-Hausdorff space, it is still probably at least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdorff.
In contrast, non-preregular spaces are encountered much more frequently in abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...

and algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

, in particular as the Zariski topology
In algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commut ...

on an algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematic ...

or the spectrum of a ring
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...

. They also arise in the model theory
In mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...

of intuitionistic logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of ...

: every complete Heyting algebra
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Arend Heyting (; 9 May 1898 – 9 July 1980) was a Netherlands, Dutch mathematician and logician.
Biography
Heyting was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic log ...

is the algebra of open set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s of some topological space, but this space need not be preregular, much less Hausdorff, and in fact usually is neither. The related concept of Scott domain
In the mathematics, mathematical fields of order theory, order and domain theory, a Scott domain is an algebraic poset, algebraic, bounded complete, bounded-complete complete partial order, cpo. They are named in honour of Dana S. Scott, who was the ...

also consists of non-preregular spaces.
While the existence of unique limits for convergent nets and filters implies that a space is Hausdorff, there are non-Hausdorff TProperties

Subspaces andproducts
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produc ...

of Hausdorff spaces are Hausdorff, but quotient spaces of Hausdorff spaces need not be Hausdorff. In fact, ''every'' topological space can be realized as the quotient of some Hausdorff space.
Hausdorff spaces are TSober spaceIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

although the converse is in general not true.
Another nice property of Hausdorff spaces is that compact set
In mathematics, more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed set, closed (i.e., containing all its limit points) and bounded set, bounded (i.e., having all ...

s are always closed. For non-Hausdorff spaces, it can be that all compact sets are closed sets (for example, the cocountable topology on an uncountable set) or not (for example, the cofinite topology on an infinite set and the Sierpiński space).
The definition of a Hausdorff space says that points can be separated by neighborhoods. It turns out that this implies something which is seemingly stronger: in a Hausdorff space every pair of disjoint compact sets can also be separated by neighborhoods, in other words there is a neighborhood of one set and a neighborhood of the other, such that the two neighborhoods are disjoint. This is an example of the general rule that compact sets often behave like points.
Compactness conditions together with preregularity often imply stronger separation axioms. For example, any locally compact In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

preregular space is completely regular
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms.
Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, w ...

. Compact
Compact as used in politics may refer broadly to a pact
A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations
International relations (IR), international affairs (IA) or internationa ...

preregular spaces are , meaning that they satisfy Urysohn's lemma
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...

and the Tietze extension theorem
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...

and have partitions of unity subordinate to locally finite open cover
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s. The Hausdorff versions of these statements are: every locally compact Hausdorff space is TychonoffTikhonov (russian: Ти́хонов, link=no; masculine), sometimes spelled as Tychonoff, or Tikhonova (; feminine) is a Russian language, Russian surname that is derived from the male given name Tikhon, the Russian form of the Greek name Τύχων ...

, and every compact Hausdorff space is normal Hausdorff.
The following results are some technical properties regarding maps (continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ga ...

and otherwise) to and from Hausdorff spaces.
Let ''$f:\; X\; \backslash to\; Y$'' be a continuous function and suppose $Y$ is Hausdorff. Then the graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...

of ''$f$'', $\backslash $, is a closed subset of ''$X\; \backslash times\; Y$''.
Let ''$f:\; X\; \backslash to\; Y$'' be a function and let $\backslash operatorname(f)\; \backslash triangleq\; \backslash $ be its kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...

regarded as a subspace of ''$X\; \backslash times\; X$''.
*If ''$f$'' is continuous and ''$Y$'' is Hausdorff then ''$\backslash ker(f)$'' is closed.
*If ''$f$'' is an open
Open or OPEN may refer to:
Music
* Open (band)
Open is a band.
Background
Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australia ...

surjection
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

and ''$\backslash ker(f)$'' is closed then ''$Y$'' is Hausdorff.
*If ''$f$'' is a continuous, open surjection (i.e. an open quotient map) then ''$Y$'' is Hausdorff if and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...

''$\backslash ker(f)$'' is closed.
If ''$f,\; g\; :\; X\; \backslash to\; Y$'' are continuous maps and ''$Y$'' is Hausdorff then the equalizer $\backslash mbox(f,g)\; =\; \backslash $ is closed in ''$X$''. It follows that if ''$Y$'' is Hausdorff and ''$f$'' and ''$g$'' agree on a dense
The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass
Mass is both a property
Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...

subset of ''$X$'' then ''$f\; =\; g$''. In other words, continuous functions into Hausdorff spaces are determined by their values on dense subsets.
Let ''$f:\; X\; \backslash to\; Y$'' be a closed surjection such that ''$f^\; (y)$'' is compact
Compact as used in politics may refer broadly to a pact
A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations
International relations (IR), international affairs (IA) or internationa ...

for all ''$y\; \backslash in\; Y$''. Then if ''$X$'' is Hausdorff so is ''$Y$''.
Let ''$f:\; X\; \backslash to\; Y$'' be a quotient map
as the quotient space of a Disk (mathematics), disk, by ''gluing'' together to a single point the points (in blue) of the boundary of the disk.
In topology and related areas of mathematics, the quotient space of a topological space under a given e ...

with ''$X$'' a compact Hausdorff space. Then the following are equivalent:
*''$Y$'' is Hausdorff.
*''$f$'' is a closed map
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

.
*''$\backslash ker(f)$'' is closed.
Preregularity versus regularity

Allregular space
In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' admit non-overlapping open neighborhoods. Thus ''p'' and ''C'' can be ...

s are preregular, as are all Hausdorff spaces. There are many results for topological spaces that hold for both regular and Hausdorff spaces.
Most of the time, these results hold for all preregular spaces; they were listed for regular and Hausdorff spaces separately because the idea of preregular spaces came later.
On the other hand, those results that are truly about regularity generally do not also apply to nonregular Hausdorff spaces.
There are many situations where another condition of topological spaces (such as paracompactness or local compactness) will imply regularity if preregularity is satisfied.
Such conditions often come in two versions: a regular version and a Hausdorff version.
Although Hausdorff spaces are not, in general, regular, a Hausdorff space that is also (say) locally compact will be regular, because any Hausdorff space is preregular.
Thus from a certain point of view, it is really preregularity, rather than regularity, that matters in these situations.
However, definitions are usually still phrased in terms of regularity, since this condition is better known than preregularity.
See History of the separation axioms
The history of the separation axioms in general topology has been convoluted, with many meanings competing for the same terms and many terms competing for the same concept.
Origins
Before the current general definition of topological space, the ...

for more on this issue.
Variants

The terms "Hausdorff", "separated", and "preregular" can also be applied to such variants on topological spaces asuniform space
In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s, Cauchy space In general topology and mathematical analysis, analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by H. H. Keller in 1968, as ...

s, and convergence space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s.
The characteristic that unites the concept in all of these examples is that limits of nets and filters (when they exist) are unique (for separated spaces) or unique up to topological indistinguishability (for preregular spaces).
As it turns out, uniform spaces, and more generally Cauchy spaces, are always preregular, so the Hausdorff condition in these cases reduces to the TAlgebra of functions

The algebra of continuous (real or complex) functions on a compact Hausdorff space is a commutativeC*-algebra
In mathematics, specifically in functional analysis
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, and conversely by the Banach–Stone theorem one can recover the topology of the space from the algebraic properties of its algebra of continuous functions. This leads to noncommutative geometryNoncommutative geometry (NCG) is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, chang ...

, where one considers noncommutative C*-algebras as representing algebras of functions on a noncommutative space.
Academic humour

* Hausdorff condition is illustrated by the pun that in Hausdorff spaces any two points can be "housed off" from each other by open sets.Colin Adams and Robert Franzosa. ''Introduction to Topology: Pure and Applied.'' p. 42 * In the Mathematics Institute of theUniversity of Bonn
The Rhenish Friedrich Wilhelm University of Bonn (german: Rheinische Friedrich-Wilhelms-Universität Bonn) is a public research university
A public university or public college is a university
A university ( la, universitas, 'a whole') is ...

, in which Felix Hausdorff
Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...

researched and lectured, there is a certain room designated the Hausdorff-Raum. This is a pun, as ''Raum'' means both ''room'' and ''space'' in German.
See also

* * * , a Hausdorff space ''X'' such that every continuous function has a fixed point.Notes

References

* Arkhangelskii, A.V., L.S. Pontryagin, ''General Topology I'', (1990) Springer-Verlag, Berlin. . * Bourbaki; ''Elements of Mathematics: General Topology'', Addison-Wesley (1966). * * {{DEFAULTSORT:Hausdorff Space Separation axioms Properties of topological spaces