In topology and related branches of mathematics, a Hausdorff space, separated space or T₂ space is a topological space in which points can be separated by neighbourhoods. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. 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 In Topology and related branches of Mathematics, the T0 spaces or Kolmogorov spaces, named after Andrey Kolmogorov, form a broad class In Topology and related branches of Mathematics, T1 spaces and R0 spaces are particular kinds of Topological spaces The In Topology, completely Hausdorff spaces and Urysohn (or T2½) spaces are types of Topological spaces satisfying slightly In Topology, completely Hausdorff spaces and Urysohn (or T2½) spaces are types of Topological spaces satisfying slightly In Topology and related fields of Mathematics, regular spaces and T3 spaces are particularly convenient kinds of Topological spaces In Topology and related branches of Mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of Topological spaces In Topology and related branches of Mathematics, normal spaces, T4 spaces, T5 spaces, and T6 spaces In Topology and related branches of Mathematics, normal spaces, T4 spaces, T5 spaces, and T6 spaces In Topology and related branches of Mathematics, normal spaces, T4 spaces, T5 spaces, and T6 spaces Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently used and discussed. 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 It implies the uniqueness of limits of sequences, nets, and filters. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" The limit of a sequence is one of the oldest concepts in Mathematical analysis. This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics In Mathematics, a filter is a special Subset of a Partially ordered set.

Hausdorff spaces are named for Felix Hausdorff, one of the founders of topology. Felix Hausdorff ( November 8, 1868 &ndash January 26, 1942) was a German Mathematician who is considered to be one of the founders Hausdorff's original definition of a topological space included the Hausdorff condition as an axiom.

Definitions

The points x and y, separated by their respective neighbourhoods U and V.

Suppose that X is a topological space. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Let x and y be points in X. In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume We say that x and y 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 (U ∩ V = ). In Topology and related branches of Mathematics, separated sets are pairs of Subsets of a given Topological space that are related to each other In Predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. X is a Hausdorff space if any two distinct points of X can be separated by neighborhoods. Two or more things are distinct if no two of them are the same thing This is why Hausdorff spaces are also called T₂ spaces or separated spaces.

X is a preregular space if any two topologically distinguishable points can be separated by neighbourhoods. In Topology, two points of a Topological space X are topologically indistinguishable if they have exactly the same neighborhoods That is if Preregular spaces are also called R₁ spaces.

The relationship between these two conditions is as follows. A topological space is Hausdorff if and only if it is both preregular and Kolmogorov (i. ↔ In Topology and related branches of Mathematics, the T0 spaces or Kolmogorov spaces, named after Andrey Kolmogorov, form a broad class e. distinct points are topologically distinguishable). A topological space is preregular if and only if its Kolmogorov quotient is Hausdorff. In Topology and related branches of Mathematics, the T0 spaces or Kolmogorov spaces, named after Andrey Kolmogorov, form a broad class

Equivalences

For a topological space X, the following are equivalent:

• X is Hausdorff space.
• Limits in X are unique (i. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" e. sequences, nets and filters converge to at most one point). The limit of a sequence is one of the oldest concepts in Mathematical analysis. This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics In Mathematics, a filter is a special Subset of a Partially ordered set.
• Every singleton set contained in X is equal to the intersection of all closed neighbourhoods containing it.
• The diagonal Δ = {(x,x) | x ∈ X} is closed as a subset of the product space X × X. In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural

Examples and counterexamples

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. Analysis has its beginnings in the rigorous formulation of Calculus. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, the real numbers may be described informally in several different ways More generally, all metric spaces are Hausdorff. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In fact, many spaces of use in analysis, such as topological groups and topological manifolds, have the Hausdorff condition explicitly stated in their definitions. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the In Mathematics, a topological manifold is a Hausdorff Topological space which looks locally like Euclidean space in a sense defined below

A simple example of a topology that is T₁ but is not Hausdorff is the cofinite topology. In Topology and related branches of Mathematics, T1 spaces and R0 spaces are particular kinds of Topological spaces The In Mathematics, a cofinite Subset of a set X is a subset Y whose complement in X is a finite 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. In Mathematics, a pseudometric space is a generalized Metric space in which the distance between two distinct points can be zero In Topology and related areas of Mathematics a gauge space is a Topological space where the Topology is defined by a family of 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. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Mathematics, namely Algebraic geometry, the Zariski topology is a particular Topology chosen for algebraic varieties that reflects the algebraic This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of 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. In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models Intuitionistic logic, or constructivist logic, is the Symbolic logic system originally developed by Arend Heyting to provide a formal basis for Brouwer In Mathematics, a complete lattice is a Partially ordered set in which all subsets have both a Supremum (join and an Infimum (meet In Mathematics, Heyting algebras are special Partially ordered sets that constitute a generalization of Boolean algebras named after Arend Heyting In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in

Properties

Subspaces and products of Hausdorff spaces are Hausdorff,^[1] but quotient spaces of Hausdorff spaces need not be Hausdorff. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying 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. In Topology and related branches of Mathematics, T1 spaces and R0 spaces are particular kinds of Topological spaces The In Mathematics, a singleton is a set with exactly one element Similarly, preregular spaces are R₀. In Topology and related branches of Mathematics, T1 spaces and R0 spaces are particular kinds of Topological spaces The

Another nice property of Hausdorff spaces is that compact sets are always closed. ^[2] This may fail for spaces which are non-Hausdorff (there are examples of T₁ spaces where it fails).

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 be separated by neighborhoods. ^[3] 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. In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks In Topology and related branches of Mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of Topological spaces Compact 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 covers. In Topology and related branches of Mathematics, normal spaces, T4 spaces, T5 spaces, and T6 spaces In Topology, Urysohn's lemma, sometimes called "the first non-trivial fact of point set topology" is commonly used to construct continuous functions of In Topology, the Tietze extension theorem states that if X is a Normal topological space and f: A &rarr R In Mathematics, a partition of unity of a Topological space X is a set of continuous functions \{\rho_i\}_{i\in I} from X In Mathematics, a cover of a set X is a collection of sets such that X is a Subset of the union of sets in the collection The Hausdorff versions of these statements are: every locally compact Hausdorff space is Tychonoff, and every compact Hausdorff space is normal Hausdorff. In Topology and related branches of Mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of Topological spaces

The following results are some technical properties regarding maps (continuous and otherwise) to and from Hausdorff spaces. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function

Let f : X → Y be a continuous function and suppose Y is Hausdorff. Then the graph of f, , is a closed subset of X × Y. In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x)

Let f : X → Y be a function and let be its kernel regarded as a subspace of X × X. In Mathematics, the kernel of a function f may be taken to be either the Equivalence relation on the function's domain

• If f is continuous and Y is Hausdorff then ker(f) is closed.
• If f is an open surjection and ker(f) is closed then Y is Hausdorff. In Topology, an open map is a function between two Topological spaces which maps Open sets to open sets In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every
• If f is a continuous, open surjection (i. e. an open quotient map) then Y is Hausdorff if and only if ker(f) is closed. ↔

If f,g : X → Y are continuous maps and Y is Hausdorff then the equalizer is closed in X. In Mathematics, an equaliser, or equalizer, is a set of arguments where two or more functions have equal values It follows that if Y is Hausdorff and f and g agree on a dense subset of X then f = g. In Topology and related areas of Mathematics, a Subset A of a Topological space X is called dense (in X) if In other words, continuous functions into Hausdorff spaces are determined by their values on dense subsets.

Let f : X → Y be a closed surjection such that f⁻¹(y) is compact for all y ∈ Y. In Topology, an open map is a function between two Topological spaces which maps Open sets to open sets Then if X is Hausdorff so is Y.

Let f : X → Y be a quotient map with X a compact Hausdorff space. In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying Then the following are equivalent

• Y is Hausdorff
• f is a closed map
• ker(f) is closed

Preregularity versus regularity

All regular spaces are preregular, as are all Hausdorff spaces. In Topology, an open map is a function between two Topological spaces which maps Open sets to open sets In Topology and related fields of Mathematics, regular spaces and T3 spaces are particularly convenient kinds of Topological 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 don't 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. In Mathematics, a paracompact space is a Topological space in which every Open cover admits an open locally finite refinement. In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks Such conditions often come in two versions: a regular version and a Hausdorff version. Although Hausdorff spaces aren't generally 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 more well known than preregularity.

See History of the separation axioms for more on this issue. In General topology, the Separation axioms have had a convoluted history with many competing meanings for the same term and many competing terms for the same concept

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. In the Mathematical field of Topology, a uniform space is a set with a uniform structure. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has 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. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has 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).

Notes

References

• Arkhangelskii, A. PlanetMath is a free, collaborative online Mathematics Encyclopedia. PlanetMath is a free, collaborative online Mathematics Encyclopedia. PlanetMath is a free, collaborative online Mathematics Encyclopedia. V. , L. S. Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4
• Bourbaki; Elements of Mathematics: General Topology, Addison-Wesley (1966). Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition
• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0486434796.