- Hausdorff space
In

topology and related branches ofmathematics , a**Hausdorff space**,**separated space**or**T**is a_{2}spacetopological space in which distinct points have disjoint neighbourhoods. Of the manyseparation axiom s that can be imposed on a topological space, the "Hausdorff condition" (T_{2}) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Intuitively, the condition is illustrated by the pun that a space is Hausdorff if any two points can be "housed off" from each other byopen sets .Hausdorff spaces are named for

Felix Hausdorff , one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom.**Definitions**Suppose that "X" is a

topological space . Let "x" and "y" be points in "X". We say that "x" and "y" can be "separated by neighbourhoods " if there exists a neighbourhood "U" of "x" and a neighbourhood "V" of "y" such that "U" and "V" are disjoint ("U" ∩ "V" = $varnothing$)."X" is a**Hausdorff space**if any twodistinct points of "X" can be separated by neighborhoods. This condition is the thirdseparation axiom (after T_{0}and 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 twotopologically distinguishable points can be separated by 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 it is both preregular (i.e. topologically distinguishable points are separated) and Kolmogorov (i.e. distinct points are topologically distinguishable). A topological space is preregular if and only if itsKolmogorov quotient is Hausdorff.**Equivalences**For a topological space "X", the following are equivalent:

* "X" is Hausdorff space.

* Limits in "X" are unique (i.e. sequences, nets and filters converge to at most one point).

* Everysingleton set contained in "X" is equal to the intersection of all closed neighbourhoods containing it.

* The diagonal Δ = {("x","x") | "x" ∈ "X"} is closed as a subset of theproduct space "X" × "X".**Examples and counterexamples**Almost all spaces encountered in analysis are Hausdorff; most importantly, the

real number s (under the standardmetric topology on real numbers) are a Hausdorff space. More generally, allmetric space s are Hausdorff. In fact, many spaces of use in analysis, such astopological group s andtopological manifold s, have the Hausdorff condition explicitly stated in their definitions.A simple example of a topology that is T

_{1}but is not Hausdorff is thecofinite topology .Pseudometric space s typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in the construction of Hausdorffgauge space 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 andalgebraic geometry , in particular as theZariski topology on analgebraic variety or thespectrum of a ring . They also arise in themodel theory ofintuitionistic logic : every completeHeyting algebra is the algebra ofopen set s of some topological space, but this space need not be preregular, much less Hausdorff.**Properties**Subspaces and products of Hausdorff spaces are Hausdorff, [

*planetmath reference|id=7202|title=Hausdorff property is hereditary*] butquotient space s 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 T

_{1}, meaning that all singletons are closed. Similarly, preregular spaces are R_{0}.Another nice property of Hausdorff spaces is that

compact set s are always closed. [*planetmath reference|id=4203|title=Proof of A compact set in a Hausdorff space is closed*] This may fail for spaces which are non-Hausdorff (there are examples of T_{1}spaces where it fails).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 [

*planetmath reference|id=4193|title=Point and a compact set in a Hausdorff space have disjoint open 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 preregular space is completely regular. Compact preregular spaces are normal, meaning that they satisfy

Urysohn's lemma and theTietze extension theorem and have partitions of unity subordinate to locally finiteopen cover s. The Hausdorff versions of these statements are: every locally compact Hausdorff space is Tychonoff, and every compact Hausdorff space is normal Hausdorff.The following results are some technical properties regarding maps (continuous and otherwise) to and from Hausdorff spaces.

Let "f" : "X" → "Y" be a continuous function and suppose "Y" is Hausdorff. Then the graph of "f", $\{(x,f(x))\; mid\; xin\; X\}$, is a closed subset of "X" × "Y".

Let "f" : "X" → "Y" be a function and let $mbox\{ker\}(f)\; =\; \{(x,x\text{'})\; mid\; f(x)\; =\; f(x\text{'})\}$ be its kernel regarded as a subspace of "X" × "X".

*If "f" is continuous and "Y" is Hausdorff then ker("f") is closed.

*If "f" is an opensurjection and ker("f") is closed then "Y" is Hausdorff.

*If "f" is a continuous, open surjection (i.e. an open quotient map) then "Y" is Hausdorffif and only if ker(f) is closed.If "f,g" : "X" → "Y" are continuous maps and "Y" is Hausdorff then the equalizer $mbox\{eq\}(f,g)\; =\; \{x\; mid\; f(x)\; =\; g(x)\}$ is closed in "X". It follows that if "Y" is Hausdorff and "f" and "g" agree on a dense 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" → "Y" be a closed surjection such that "f"

^{−1}("y") is compact for all "y" ∈ "Y". Then if "X" is Hausdorff so is "Y".Let "f" : "X" → "Y" be a

quotient map with "X" a compact Hausdorff space. Then the following are equivalent

*"Y" is Hausdorff

*"f" is aclosed map

*ker("f") is closed**Preregularity versus regularity**All

regular space 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 don't also apply to nonregular Hausdorff spaces.There are many situations where another condition of topological spaces (such as

paracompactness orlocal 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 aren't generally 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 for more on this issue.**Variants**The terms "Hausdorff", "separated", and "preregular" can also be applied to such variants on topological spaces as

uniform space s, Cauchy spaces, andconvergence space 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 T

_{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).**Notes****References*** Arkhangelskii, A.V.,

L.S. Pontryagin , "General Topology I", (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4

* Bourbaki; "Elements of Mathematics: General Topology", Addison-Wesley (1966).

*cite book | author=Willard, Stephen | title=General Topology | publisher=Dover Publications | year=2004 | id=ISBN 0486434796**See also***

Weak Hausdorff space

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**Hausdorff space**— /hows dawrf, howz /, Math. a topological space in which each pair of points can be separated by two disjoint open sets containing the points. [named after Felix Hausdorff (1868 1942), German mathematician, who first described it] * * * ▪… … Universalium**Hausdorff space**— /hows dawrf, howz /, Math. a topological space in which each pair of points can be separated by two disjoint open sets containing the points. [named after Felix Hausdorff (1868 1942), German mathematician, who first described it] … Useful english dictionary**Hausdorff space**— noun A topological space in which for any two distinct points x and y, there is a pair of disjoint open sets U and V such that and . Syn: T2 space (T<small>2</small> space) … Wiktionary**Completely Hausdorff space**— Separation Axioms in Topological Spaces Kolmogorov (T0) version T0 | T1 | T2 | T2½ | completely T2 T3 | T3½ | T4 | T5 | T6 In topology, an Urysohn space, or T2½ spac … Wikipedia**Locally Hausdorff space**— In mathematics, in the field of topology, a topological space is said to be locally Hausdorff if every point has an open neighbourhood that is Hausdorff under the subspace topology.Here are some facts:* Every Hausdorff space is locally Hausdorff … Wikipedia**Weak Hausdorff space**— In mathematics, a weak Hausdorff space is a topological space where the image of every continuous map from a compact Hausdorff space into the space is closed. [cite web |url=http://neil strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf |title … Wikipedia**Continuous functions on a compact Hausdorff space**— In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space with values in the real or complex numbers. This space, denoted by C(X), is a vector… … Wikipedia**Collectionwise Hausdorff space**— In mathematics, in the field of topology, a topological space is said to be collectionwise Hausdorff if given any closed discrete collection of points in the topological space, there are pairwise disjoint open sets containing the points. Every T1 … Wikipedia**Hausdorff**— may refer to:* A Hausdorff space, when used as an adjective, as in the real line is Hausdorff. * Felix Hausdorff, the German mathematician that Hausdorff spaces are named after. * Hausdorff dimension, a measure theoretic concept of dimension. *… … Wikipedia**Hausdorff-Trennungsaxiom**— Hausdorff Raum (T2) berührt die Spezialgebiete Mathematik Topologie ist Spezialfall von topologischer Raum präregulärer Raum ( … Deutsch Wikipedia