Hausdorff space
Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed.It implies the uniqueness of limits of sequences, nets, and filters.A related, but weaker, notion is that of a preregular space.is a preregular space if any two topologically distinguishable points can be separated by disjoint neighbourhoods.A topological space is preregular if and only if its Kolmogorov quotient is Hausdorff., the following are equivalent:[2] Almost all spaces encountered in analysis are Hausdorff; most importantly, the real numbers (under the standard metric topology on real numbers) are a Hausdorff space.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.They also arise in the model theory of intuitionistic logic: every complete Heyting algebra is the algebra of open sets 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 T1 spaces in which every convergent sequence has a unique limit.[7] For sequential spaces, this notion is equivalent to being weakly Hausdorff.[8] Hausdorff spaces are T1, meaning that each singleton is a closed set.The definition of a Hausdorff space says that points can be separated by neighborhoods.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.[10][11] Compact preregular spaces are normal,[12] meaning that they satisfy Urysohn's lemma and the Tietze extension theorem and have partitions of unity subordinate to locally finite open covers.The following results are some technical properties regarding maps (continuous and otherwise) to and from Hausdorff spaces.In other words, continuous functions into Hausdorff spaces are determined by their values on dense subsets.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.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.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.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.
Separation axiomstopological spacesKolmogorovcompletely T2Historytopologymathematicstopological spacedisjointneighbourhoodslimitssequencesfiltersFelix Hausdorffseparated by neighbourhoodsthere existsneighbourhoodseparation axiomtopologically distinguishableif and only ifKolmogorov quotientsingleton setclosed neighbourhoodsclosed setclosedproduct spacelifting propertyNon-Hausdorff manifoldanalysisreal numbersmetric topologymetric spacestopological groupstopological manifoldscofinite topologyinfinite setcocountable topologyuncountable setPseudometric spacesgauge spacesabstract algebraalgebraic geometryZariski topologyalgebraic varietyspectrum of a ringmodel theoryintuitionistic logiccompleteHeyting algebraopen setsScott domainsequential spacesweakly HausdorffSubspacesproductsquotient spacessingletonSober spacecompact setSierpiński spacelocally compactcompletely regularCompactnormalUrysohn's lemmaTietze extension theorempartitions of unityopen coversTychonoffcontinuouskernelsurjectionequalizerquotient mapclosed mapregular spacesparacompactnesslocal compactnessHistory of the separation axiomsuniform spacesCauchy spacesconvergence spacescompletenessC*-algebraBanach–Stone theoremnoncommutative geometryUniversity of BonnFixed-point spaceLocally Hausdorff spaceQuasitopological spaceWeak Hausdorff spaceBanach–Mazur compactumTopology and Its ApplicationsThe American Mathematical MonthlyQuarterly Journal of MathematicsAdams, ColinPearson Prentice HallPontryagin, L.S.SpringerBourbakiAddison-WesleyEncyclopedia of MathematicsEMS PressSchechter, EricDover PublicationsGeneral (point-set)AlgebraicCombinatorialContinuumDifferentialGeometriclow-dimensionalHomologycohomologySet-theoreticDigitalOpen setInteriorContinuityconnectedmetricuniformHomotopyhomotopy groupfundamental groupSimplicial complex