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In topology and related branches of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many
separation axiom In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometim ...
s 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 $x$ and $y$ in a topological space $X$ can be '' separated by neighbourhoods'' if there exists a neighbourhood $U$ of $x$ and a neighbourhood $V$ of $y$ such that $U$ and $V$ are disjoint $\left(U\cap V=\varnothing\right)$. $X$ is a Hausdorff space if any two distinct points in $X$ are separated by neighbourhoods. This condition is the third
separation axiom In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometim ...
(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. $X$ 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 ''$X$'', the following are equivalent: * $X$ is a Hausdorff space. * Limits of nets in ''$X$'' are unique. * Limits of
filters Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component that ...
on ''$X$'' are unique. * Any singleton set $\ \subset X$ is equal to the intersection of all closed neighbourhoods of ''$x$''. (A closed neighbourhood of ''$x$'' is a closed set that contains an open set containing ''x''.) * The diagonal ''$\Delta = \$'' is
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
as a subset of the product space ''$X \times X$''. * Any injection from the discrete space with two points to ''$X$'' 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

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. 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 In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
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 Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
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''.

# Properties

Subspaces and products of Hausdorff spaces are Hausdorff, but quotient spaces of Hausdorff spaces need not be Hausdorff. In fact, ''every'' topological space can be realized as the quotient of some Hausdorff space. Hausdorff spaces are T1, meaning that all singletons are closed. Similarly, preregular spaces are R0. Every Hausdorff space is a Sober space although the converse is in general not true. Another nice property of Hausdorff spaces is that compact sets are always closed. For non-Hausdorff spaces, it can be that all compact sets are closed sets (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, 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 preregular space is completely regular.
Compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
preregular spaces are normal, meaning that they satisfy Urysohn's lemma and the Tietze extension theorem and have partitions of unity subordinate to locally finite
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alp ...
s. The Hausdorff versions of these statements are: every locally compact Hausdorff space is Tychonoff, and every compact Hausdorff space is normal Hausdorff. The following results are some technical properties regarding maps (
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
and otherwise) to and from Hausdorff spaces. Let ''$f: X \to Y$'' be a continuous function and suppose $Y$ is Hausdorff. Then the graph of ''$f$'', $\$, is a closed subset of ''$X \times Y$''. Let ''$f: X \to Y$'' be a function and let $\operatorname\left(f\right) \triangleq \$ be its kernel regarded as a subspace of ''$X \times X$''. *If ''$f$'' is continuous and ''$Y$'' is Hausdorff then ''$\ker\left(f\right)$'' is closed. *If ''$f$'' is an
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * Open (Blues Image album), ''Open'' (Blues Image album), 1969 * Open (Gotthard album), ''Open'' (Gotthard album), 1999 * Open (C ...
surjection and ''$\ker\left(f\right)$'' is closed then ''$Y$'' is Hausdorff. *If ''$f$'' is a continuous, open surjection (i.e. an open quotient map) then ''$Y$'' is Hausdorff if and only if ''$\ker\left(f\right)$'' is closed. If ''$f, g : X \to Y$'' are continuous maps and ''$Y$'' is Hausdorff then the equalizer $\mbox\left(f,g\right) = \$ is closed in ''$X$''. It follows that if ''$Y$'' is Hausdorff and ''$f$'' and ''$g$'' agree on a dense subset of ''$X$'' then ''$f = g$''. In other words, continuous functions into Hausdorff spaces are determined by their values on dense subsets. Let ''$f: X \to Y$'' be a
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
surjection such that ''$f^ \left(y\right)$'' is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
for all ''$y \in Y$''. Then if ''$X$'' is Hausdorff so is ''$Y$''. Let ''$f: X \to Y$'' be a quotient map with ''$X$'' a compact Hausdorff space. Then the following are equivalent: *''$Y$'' is Hausdorff. *''$f$'' is a
closed map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, ...
. *''$\ker\left(f\right)$'' is closed.

# Preregularity versus regularity

All
regular space In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' admit non-overlapping open neighborhoods. Thus ''p'' and ''C'' ca ...
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 spaces, Cauchy spaces, and convergence 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 which 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.