| Separation axioms intopological spaces | |
|---|---|
| Kolmogorov classification | |
| T0 | (Kolmogorov) |
| T1 | (Fréchet) |
| T2 | (Hausdorff) |
| T2½ | (Urysohn) |
| completely T2 | (completely Hausdorff) |
| T3 | (regular Hausdorff) |
| T3½ | (Tychonoff) |
| T4 | (normal Hausdorff) |
| T5 | (completely normal Hausdorff) |
| T6 | (perfectly normal Hausdorff) |
Intopology and related branches ofmathematics, aHausdorff space (/ˈhaʊsdɔːrf/HOWSS-dorf,/ˈhaʊzdɔːrf/HOWZ-dorf[1]),T2 space orseparated space, is atopological space where distinct points havedisjointneighbourhoods. Of the manyseparation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness oflimits ofsequences,nets, andfilters.[2]
Hausdorff spaces are named afterFelix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as anaxiom.[3]
Points and in a topological space can beseparated by neighbourhoods ifthere exists aneighbourhood of and a neighbourhood of such that and aredisjoint. is aHausdorff space if any two distinct points in are separated by neighbourhoods. This condition is the thirdseparation axiom (after T0 and T1), which is why Hausdorff spaces are also calledT2 spaces. The nameseparated space is also used.
A related, but weaker, notion is that of apreregular space. is a preregular space if any twotopologically distinguishable points can be separated by disjoint neighbourhoods. A preregular space is also called anR1 space.
The relationship between these two conditions is as follows. A topological space is Hausdorffif and only if it is both preregular (i.e. topologically distinguishable points are separated by neighbourhoods) andKolmogorov (i.e. distinct points are topologically distinguishable). A topological space is preregular if and only if itsKolmogorov quotient is Hausdorff.
For a topological space, the following are equivalent:[2]
Almost all spaces encountered inanalysis are Hausdorff; most importantly, thereal numbers (under the standardmetric topology on real numbers) are a Hausdorff space. More generally, allmetric spaces are Hausdorff. In fact, many spaces of use in analysis, such astopological groups andtopological manifolds, have the Hausdorff condition explicitly stated in their definitions.
A simple example of a topology that isT1 but is not Hausdorff is thecofinite topology defined on aninfinite set, as is thecocountable topology defined on anuncountable set.
Pseudometric spaces typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in the construction of Hausdorffgauge spaces. 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.[6]
In contrast, non-preregular spaces are encountered much more frequently inabstract algebra andalgebraic geometry, in particular as theZariski topology on analgebraic variety or thespectrum of a ring. They also arise in themodel theory ofintuitionistic logic: everycompleteHeyting algebra is the algebra ofopen sets of some topological space, but this space need not be preregular, much less Hausdorff, and in fact usually is neither. The related concept ofScott 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 T1 spaces in which every convergent sequence has a unique limit.[7] Such spaces are calledUS spaces.[8] Forsequential spaces, this notion is equivalent to beingweakly Hausdorff.
Subspaces andproducts of Hausdorff spaces are Hausdorff, butquotient spaces of Hausdorff spaces need not be Hausdorff. In fact,every topological space can be realized as the quotient of some Hausdorff space.[9]
Hausdorff spaces areT1, meaning that eachsingleton is a closed set. Similarly, preregular spaces areR0. Every Hausdorff space is aSober space although the converse is in general not true.
Another property of Hausdorff spaces is that eachcompact set is a closed set. For non-Hausdorff spaces, it can be that each compact set is a closed set (for example, thecocountable topology on an uncountable set) or not (for example, thecofinite topology on an infinite set and theSierpiński 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,[10] 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, anylocally compact preregular space iscompletely regular.[11][12]Compact preregular spaces arenormal,[13] meaning that they satisfyUrysohn's lemma and theTietze extension theorem and havepartitions of unity subordinate to locally finiteopen covers. The Hausdorff versions of these statements are: every locally compact Hausdorff space isTychonoff, 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 be a continuous function and suppose is Hausdorff. Then thegraph of,, is a closed subset of.
Let be a function and let be itskernel regarded as a subspace of.
If are continuous maps and is Hausdorff then theequalizer is a closed set in. It follows that if is Hausdorff and and agree on adense subset of then. In other words, continuous functions into Hausdorff spaces are determined by their values on dense subsets.
Let be aclosed surjection such that iscompact for all. Then if is Hausdorff so is.
Let be aquotient map with a compact Hausdorff space. Then the following are equivalent:
Allregular spaces 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 asparacompactness 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 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.
SeeHistory of the separation axioms for more on this issue.
The terms "Hausdorff", "separated", and "preregular" can also be applied to such variants on topological spaces asuniform spaces,Cauchy spaces, andconvergence spaces. 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 T0 condition. These are also the spaces in whichcompleteness 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 atleast one limit, while a space is Hausdorff if and only if every Cauchy net has atmost one limit (since only Cauchy nets can have limits in the first place).
The algebra of continuous (real or complex) functions on a compact Hausdorff space is a commutativeC*-algebra, and conversely by theBanach–Stone theorem one can recover the topology of the space from the algebraic properties of its algebra of continuous functions. This leads tononcommutative geometry, where one considers noncommutative C*-algebras as representing algebras of functions on a noncommutative space.
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