Editing Hausdorff space

{{Short description|Type of topological space}}
{{Separation axioms}}

In [[topology]] and related branches of [[mathematics]], a '''Hausdorff space''' ({{IPAc-en|ˈ|h|aʊ|s|d|ɔːr|f}} {{Respell|HOWSS|dorf}}, {{IPAc-en|ˈ|h|aʊ|z|d|ɔːr|f}} {{Respell|HOWZ|dorf}}<ref>{{cite web |url = https://www.dictionary.com/browse/hausdorff-space |title = Hausdorff space Definition & Meaning |website=www.dictionary.com|access-date = 15 June 2022 }}</ref>), '''T<sub>2</sub> space''' or '''separated space''', is a [[topological space]] where distinct points have [[disjoint sets|disjoint]] [[neighbourhood (mathematics)|neighbourhood]]s.  Of the many [[separation axiom]]s that can be imposed on a topological space, the "Hausdorff condition" (T<sub>2</sub>) is the most frequently used and discussed. It implies the uniqueness of [[Limit of a sequence|limits]] of [[limit of a sequence|sequence]]s, [[net (topology)|net]]s, and [[filter (topology)|filter]]s.<ref name="ncatlab">{{cite web | url=https://ncatlab.org/nlab/show/separation+axioms | title=Separation axioms in nLab | website=ncatlab.org | access-date=2019-10-16 | archive-date=2020-09-30 | archive-url=https://web.archive.org/web/20200930160358/https://www.ncatlab.org/nlab/show/separation+axioms | url-status=live }}</ref>

Hausdorff spaces are named after [[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 ==
[[File:Hausdorff space.svg|thumb|203px|right|The points x and y, separated by their respective neighbourhoods U and V.]]

Points <math>x</math> and <math>y</math> in a topological space <math>X</math> can be ''[[separated by neighbourhoods]]'' if [[existential quantification|there exists]] a [[neighbourhood (topology)|neighbourhood]] <math>U</math> of <math>x</math> and a neighbourhood <math>V</math> of <math>y</math> such that <math>U</math> and <math>V</math> are [[Disjoint sets|disjoint]] <math>(U\cap V=\varnothing)</math>. <math>X</math> is a '''Hausdorff space''' if any two distinct points in <math>X</math> are separated by neighbourhoods. This condition is the third [[separation axiom]] (after T<sub>0</sub> and T<sub>1</sub>), which is why Hausdorff spaces are also called '''T<sub>2</sub> spaces'''. The name ''separated space'' is also used.

A related, but weaker, notion is that of a '''preregular space'''. <math>X</math> is a preregular space if any two [[topologically distinguishable]] points can be separated by disjoint neighbourhoods. A preregular space is also called an '''R<sub>1</sub> space'''.

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 by neighbourhoods) and [[Kolmogorov space|Kolmogorov]] (i.e. distinct points are topologically distinguishable). A topological space is preregular if and only if its [[Kolmogorov quotient]] is Hausdorff.

== Equivalences ==
For a topological space ''<math>X</math>'', the following are equivalent:<ref name="ncatlab"/>

* <math>X</math> is a Hausdorff space.
* Limits of [[net (topology)|nets]] in ''<math>X</math>'' are unique.<ref name=Willard04_86_7>{{harvnb|Willard|2004|pp=86–87}}</ref>
* Limits of [[filter (topology)|filters]] on ''<math>X</math>'' are unique.<ref name=Willard04_86_7/>
* Any [[singleton set]] <math>\{ x \} \subset X</math> is equal to the intersection of all [[neighbourhood (mathematics)|closed neighbourhoods]] of ''<math>x</math>''.<ref>{{harvnb|Bourbaki|1966|p=75}}</ref> (A closed neighbourhood of <math>x</math> is a [[closed set]] that contains an open set containing <math>x</math>.)
* The diagonal ''<math>\Delta = \{ (x, x) \mid x \in X \}</math>'' is [[closed set|closed]] as a subset of the [[product space]] ''<math>X \times X</math>''.
* Any injection from the discrete space with two points to ''<math>X</math>'' has the [[lifting property]] with respect to the map from the finite topological space with two open points and one closed point to a single point.

== Examples of Hausdorff and non-Hausdorff spaces ==
{{see also|Non-Hausdorff manifold}}

Almost all spaces encountered in [[mathematical analysis|analysis]] are Hausdorff; most importantly, the [[real number]]s (under the standard [[metric topology]] on real numbers) are a Hausdorff space. More generally, all [[metric space]]s are Hausdorff. In fact, many spaces of use in analysis, such as [[topological group]]s and [[topological manifold]]s, have the Hausdorff condition explicitly stated in their definitions.

A simple example of a topology that is [[T1 space|T<sub>1</sub>]] but is not Hausdorff is the [[cofinite topology]] defined on an [[infinite set]], as is the [[cocountable topology]] defined on an [[uncountable set]].

[[Pseudometric space]]s typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in the construction of Hausdorff [[gauge 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.<ref>See for instance [[Lp space#Lp spaces and Lebesgue integrals]], [[Banach–Mazur compactum]] etc.</ref>

In contrast, non-preregular spaces are encountered much more frequently in [[abstract algebra]] and [[algebraic geometry]], in particular as the [[Zariski topology]] on an [[algebraic variety]] or the [[spectrum of a ring]]. They also arise in the [[model theory]] of [[intuitionistic logic]]: every [[complete lattice|complete]] [[Heyting algebra]] is the algebra of [[open set]]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]] 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 T<sub>1</sub> spaces in which every convergent sequence has a unique limit.<ref>{{cite journal |last=van Douwen |first=Eric K. |title=An anti-Hausdorff Fréchet space in which convergent sequences have unique limits |journal=[[Topology and Its Applications]] |volume=51 |issue=2 |year=1993 |pages=147–158 |doi=10.1016/0166-8641(93)90147-6 |doi-access=free }}</ref> Such spaces are called ''US spaces''.<ref>{{cite journal | last = Wilansky | first = Albert | title = Between T<sub>1</sub> and T<sub>2</sub> | journal = [[The American Mathematical Monthly]] | volume = 74 | issue = 3 | year = 1967 | pages = 261–266 | doi = 10.2307/2316017 | jstor = 2316017 }}</ref> For [[sequential space]]s, this notion is equivalent to being [[weakly Hausdorff]].

== Properties ==
[[Subspace (topology)|Subspace]]s and [[product topology|products]] of Hausdorff spaces are Hausdorff, but [[quotient space (topology)|quotient space]]s of Hausdorff spaces need not be Hausdorff. In fact, ''every'' topological space can be realized as the quotient of some Hausdorff space.<ref>{{cite journal |last=Shimrat |first=M. |title=Decomposition spaces and separation properties |journal=[[Quarterly Journal of Mathematics]] |volume=2 |year=1956 |pages=128–129 |doi= 10.1093/qmath/7.1.128}}</ref>

Hausdorff spaces are [[T1 space|T<sub>1</sub>]], meaning that each [[singleton (mathematics)|singleton]] is a closed set. Similarly, preregular spaces are [[R0 space|R<sub>0</sub>]]. Every Hausdorff space is a [[Sober space]] although the converse is in general not true. 

Another property of Hausdorff spaces is that each [[compact set]] is a closed set. For non-Hausdorff spaces, it can be that each compact set is a closed set (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,<ref>{{harvnb|Willard|2004|pp=124}}</ref> 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 space|locally compact]] preregular space is [[completely regular space|completely regular]].{{sfn|Schechter|1996|loc=17.14(d), p. 460}}<ref>{{cite web |title=Locally compact preregular spaces are completely regular |url=https://math.stackexchange.com/questions/4503299 |website=math.stackexchange.com}}</ref> [[Compact space|Compact]] preregular spaces are [[normal space|normal]],{{sfn|Schechter|1996|loc=17.7(g), p. 457}} meaning that they satisfy [[Urysohn's lemma]] and the [[Tietze extension theorem]] and have [[partition of unity|partitions of unity]] subordinate to locally finite [[open cover]]s. The Hausdorff versions of these statements are: every locally compact Hausdorff space is [[Tychonoff space|Tychonoff]], and every compact Hausdorff space is normal Hausdorff.

The following results are some technical properties regarding maps ([[continuous (topology)|continuous]] and otherwise) to and from Hausdorff spaces.

Let ''<math>f\colon X \to Y</math>'' be a continuous function and suppose <math>Y</math> is Hausdorff. Then the [[Graph of a function|graph]] of ''<math>f</math>'', <math>\{(x,f(x)) \mid x\in X\}</math>, is a closed subset of ''<math>X \times Y</math>''.

Let ''<math>f\colon X \to Y</math>'' be a function and let <math>\ker(f) \triangleq \{(x,x') \mid f(x) = f(x')\}</math> be its [[kernel of a function|kernel]] regarded as a subspace of ''<math>X \times X</math>''.
*If ''<math>f</math>'' is continuous and ''<math>Y</math>'' is Hausdorff then ''<math>\ker(f)</math>'' is a closed set.
*If ''<math>f</math>'' is an [[open map|open]] surjection and ''<math>\ker(f)</math>'' is a closed set then ''<math>Y</math>'' is Hausdorff.
*If ''<math>f</math>'' is a continuous, open [[surjection]] (i.e. an open quotient map) then ''<math>Y</math>'' is Hausdorff [[if and only if]] ''<math>\ker(f)</math>'' is a closed set.

If ''<math>f, g \colon X \to Y</math>'' are continuous maps and ''<math>Y</math>'' is Hausdorff then the [[Equaliser (mathematics)|equalizer]] <math>\mbox{eq}(f,g) = \{x \mid f(x) = g(x)\}</math> is a closed set in ''<math>X</math>''. It follows that if ''<math>Y</math>'' is Hausdorff and ''<math>f</math>'' and ''<math>g</math>'' agree on a [[dense (topology)|dense]] subset of ''<math>X</math>'' then ''<math>f = g</math>''. In other words, continuous functions into Hausdorff spaces are determined by their values on dense subsets.

Let ''<math>f\colon X \to Y</math>'' be a [[closed map|closed]] surjection such that ''<math>f^{-1} (y)</math>'' is [[compact space|compact]] for all ''<math>y \in Y</math>''. Then if ''<math>X</math>'' is Hausdorff so is ''<math>Y</math>''.

Let ''<math>f\colon X \to Y</math>'' be a [[quotient map (topology)|quotient map]] with ''<math>X</math>'' a compact Hausdorff space. Then the following are equivalent:
*''<math>Y</math>'' is Hausdorff.
*''<math>f</math>'' is a [[closed map]].
*''<math>\ker(f)</math>'' is a closed set.

== 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 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]] 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 space]]s, and [[convergence 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<sub>0</sub> condition. These are also the spaces in which [[completeness (topology)|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).

== Algebra of functions ==
The algebra of continuous (real or complex) functions on a compact Hausdorff space is a commutative [[C*-algebra]], 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 geometry]], 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]].<ref>{{cite book |first1=Colin |last1=Adams |author1-link=Colin Adams (mathematician)|first2=Robert |last2=Franzosa |title=Introduction to Topology: Pure and Applied |publisher=[[Prentice Hall|Pearson Prentice Hall]] |date=2008 |isbn=978-0-13-184869-6 |pages=42 |url=}}</ref>
* In the Mathematics Institute of the [[Universität Bonn|University of Bonn]], in which [[Felix Hausdorff]] 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==
* {{annotated link|Fixed-point space}}, a Hausdorff space ''X'' such that every continuous function {{math|''f'' : ''X'' → ''X''}} has a fixed point.
* {{annotated link|Locally Hausdorff space}}
* {{annotated link|Non-Hausdorff manifold}}
* {{annotated link|Quasitopological space}}
* {{annotated link|Separation axiom}}
* {{annotated link|Weak Hausdorff space}}

== Notes ==
{{Reflist}}

== References ==

*{{cite book |last1=Arkhangelskii |first1=A.V. |first2=L.S. |last2=Pontryagin |author2-link=L.S. Pontryagin |title=General Topology I |publisher=[[Springer Science+Business Media|Springer]] |date=1990 |isbn=3-540-18178-4 }}
*{{cite book |author1-link=Nicolas Bourbaki |author1=Bourbaki |title=Elements of Mathematics: General Topology |publisher=[[Addison-Wesley]] |date=1966 |isbn= }}
* {{Springer |title=Hausdorff space |id=p/h046730}}
* {{Schechter Handbook of Analysis and Its Foundations}} <!--{{sfn|Schechter|1996|p=}}-->
* {{cite book |last=Willard |first=Stephen |title=General Topology |publisher=[[Dover Publications]] |year=2004 |isbn=0-486-43479-6|url=https://books.google.com/books?id=-o8xJQ7Ag2cC&q=%22Hausdorff+space%22}}

{{Topology}}

{{DEFAULTSORT:Hausdorff Space}}
[[Category:Separation axioms]]
[[Category:Properties of topological spaces]]