Editing Hausdorff space (section)

== 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]].