In topology and related branches of mathematics, a Hausdorff space ( HOWSS-dorf, HOWZ-dorf), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. 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. 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
Points x {\displaystyle x} and y {\displaystyle y} in a topological space X {\displaystyle X} can be separated by neighbourhoods if there exists a neighbourhood U {\displaystyle U} of x {\displaystyle x} and a neighbourhood V {\displaystyle V} of y {\displaystyle y} such that U {\displaystyle U} and V {\displaystyle V} are disjoint ( U ∩ V = ∅ ) {\displaystyle (U\cap V=\varnothing )} . X {\displaystyle X} is a Hausdorff space if any two distinct points in X {\displaystyle X} are separated by neighbourhoods. This condition is the third separation axiom (after T0 and T1), which is why Hausdorff spaces are also called T2 spaces. The name separated space is also used. A related, but weaker, notion is that of a preregular space. X {\displaystyle X} is a preregular space if any two topologically distinguishable points can be separated by disjoint neighbourhoods. A preregular space is also called an R1 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 (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 X {\displaystyle X} , the following are equivalent:
X {\displaystyle X} is a Hausdorff space. Limits of nets in X {\displaystyle X} are unique. Limits of filters on X {\displaystyle X} are unique. Any singleton set { x } ⊂ X {\displaystyle \{x\}\subset X} is equal to the intersection of all closed neighbourhoods of x {\displaystyle x} . (A closed neighbourhood of x {\displaystyle x} is a closed set that contains an open set containing x {\displaystyle x} .) The diagonal Δ = { ( x , x ) ∣ x ∈ X } {\displaystyle \Delta =\{(x,x)\mid x\in X\}} is closed as a subset of the product space X × X {\displaystyle X\times X} . Any injection from the discrete space with two points to X {\displaystyle X} has the left 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
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. More generally, all metric spaces are Hausdorff. In fact, many spaces of use in analysis, such as topological groups and topological manifolds, have the Hausdorff condition explicitly stated in their definitions. A simple example of a topology that is T1 but is not Hausdorff is the cofinite topology defined on an infinite set, as is the cocountable topology defined on an uncountable set. Pseudometric spaces typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in the construction of Hausdorff gauge 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. 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 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. Such spaces are called US spaces. For sequential spaces, this notion is equivalent to being weakly Hausdorff.
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