In
topology and related branches of
mathematics, a Hausdorff space ( , ), separated space or T
2 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 axioms that can be imposed on a topological space, the "Hausdorff condition" (T
2) is the most frequently used and discussed. It implies the uniqueness of
limits
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
of
sequences,
net
Net or net may refer to:
Mathematics and physics
* Net (mathematics), a filter-like topological generalization of a sequence
* Net, a linear system of divisors of dimension 2
* Net (polyhedron), an arrangement of polygons that can be folded up ...
s, and
filter
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 tha ...
s.
Hausdorff spaces are named after
Felix Hausdorff
Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, an ...
, 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
and
in a topological space
can be ''
separated by neighbourhoods'' if
there exists
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, w ...
a
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
of
and a neighbourhood
of
such that
and
are
disjoint .
is a Hausdorff space if any two distinct points in
are separated by neighbourhoods. This condition is the third
separation axiom (after T
0 and T
1), which is why Hausdorff spaces are also called T
2 spaces. The name ''separated space'' is also used.
A related, but weaker, notion is that of a preregular space.
is a preregular space if any two
topologically distinguishable
In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborhoods that contain ''x'', ...
points can be separated by disjoint neighbourhoods. A preregular space is also called an R
1 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
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
(i.e. distinct points are topologically distinguishable). A topological space is preregular if and only if its
Kolmogorov quotient
In topology and related branches of mathematics, a topological space ''X'' is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of ''X'', at least one of them has a neighborhood not containing the ...
is Hausdorff.
Equivalences
For a topological space ''
'', the following are equivalent:
*
is a Hausdorff space.
* Limits of
nets in ''
'' are unique.
* Limits of
filters on ''
'' are unique.
[
* Any ]singleton set
In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0.
Properties
Within the framework of Zermelo–Fraenkel set theory, the ...
is equal to the intersection of all closed neighbourhoods of ''''. (A closed neighbourhood of '''' is a closed set that contains an open set containing ''x''.)
* The diagonal '''' is closed as a subset of the product space
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...
''''.
* Any injection from the discrete space with two points to '''' 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
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
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 group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
s and topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout math ...
s, 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
In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocoun ...
defined on an infinite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only s ...
.
Pseudometric space
In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In the same way as every normed space is a metric ...
s 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
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
on an algebraic variety or the spectrum of a ring. They also arise in the model theory of intuitionistic logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
: every complete Heyting algebra In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of '' ...
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
In the mathematical fields of order and domain theory, a Scott domain is an algebraic, bounded-complete cpo. They are named in honour of Dana S. Scott, who was the first to study these structures at the advent of domain theory. Scott domains a ...
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
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
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 singleton
Singleton may refer to:
Sciences, technology Mathematics
* Singleton (mathematics), a set with exactly one element
* Singleton field, used in conformal field theory Computing
* Singleton pattern, a design pattern that allows only one instance ...
s are closed. Similarly, preregular spaces are R0. Every Hausdorff space is a Sober space In mathematics, a sober space is a topological space ''X'' such that every (nonempty) irreducible closed subset of ''X'' is the closure of exactly one point of ''X'': that is, every irreducible closed subset has a unique generic point.
Definitio ...
although the converse is in general not true.
Another nice property of Hausdorff spaces is that compact set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
s are always closed. For non-Hausdorff spaces, it can be that all compact sets are closed sets (for example, the cocountable topology The cocountable topology or countable complement topology on any set ''X'' consists of the empty set and all cocountable subsets of ''X'', that is all sets whose complement in ''X'' is countable. It follows that the only closed subsets are ''X'' and ...
on an uncountable set) or not (for example, the cofinite topology
In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocoun ...
on an infinite set and the Sierpiński space
In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed.
It is the smallest example of a topological space which is neither trivial nor discrete. It is name ...
).
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
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
. Compact preregular spaces are normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
, meaning that they satisfy Urysohn's lemma and the Tietze extension theorem
In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that Continuous function (topology), continuous functions on a closed subset of a Normal space, normal topological space can be extend ...
and have partitions of unity
In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X:
* there is a neighbourhood of where all but a finite number of the functions of are 0, ...
subordinate to locally finite open covers. 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 and otherwise) to and from Hausdorff spaces.
Let '''' be a continuous function and suppose is Hausdorff. Then the graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
of '''', , is a closed subset of ''''.
Let '''' be a function and let be its kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
regarded as a subspace of ''''.
*If '''' is continuous and '''' is Hausdorff then '''' is closed.
*If '''' is an open surjection
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
and '''' is closed then '''' is Hausdorff.
*If '''' is a continuous, open surjection (i.e. an open quotient map) then '''' is Hausdorff if and only if '''' is closed.
If '''' are continuous maps and '''' is Hausdorff then the equalizer is closed in ''''. It follows that if '''' is Hausdorff and '''' and '''' agree on a dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
subset of '''' then ''''. In other words, continuous functions into Hausdorff spaces are determined by their values on dense subsets.
Let '''' be a closed surjection such that '''' is compact for all ''''. Then if '''' is Hausdorff so is ''''.
Let '''' be a quotient map
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
with '''' a compact Hausdorff space. Then the following are equivalent:
*'''' is Hausdorff.
*'''' is a closed map.
*'''' is closed.
Preregularity versus regularity
All regular spaces 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
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, ...
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
The history of the separation axioms in general topology has been convoluted, with many meanings competing for the same terms and many terms competing for the same concept.
Origins
Before the current general definition of topological space, th ...
for more on this issue.
Variants
The terms "Hausdorff", "separated", and "preregular" can also be applied to such variants on topological spaces as uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
s, Cauchy space In general topology and analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by H. H. Keller in 1968, as an axiomatic tool deriv ...
s, 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 In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone.
In brief, the Banach–Stone theorem allows one to recove ...
one can recover the topology of the space from the algebraic properties of its algebra of continuous functions. This leads to noncommutative geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some g ...
, where one considers noncommutative C*-algebras as representing algebras of functions on a noncommutative space.
Academic humour
* Hausdorff condition is illustrated by the pun that in Hausdorff spaces any two points can be "housed off" from each other by open sets
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
.
* In the Mathematics Institute of the University of Bonn, in which Felix Hausdorff
Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, an ...
researched and lectured, there is a certain room designated the Hausdorff-Raum. This is a pun, as ''Raum'' means both ''room'' and ''space'' in German.
See also
* , a Hausdorff space ''X'' such that every continuous function has a fixed point.
*
*
*
*
*
Notes
References
*
*
*
*
*
{{DEFAULTSORT:Hausdorff Space
Separation axioms
Properties of topological spaces