and related branches of 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
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
s, and filter
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
in a topological space
can be ''separated by neighbourhoods
'' if there exists
and a neighbourhood
is a Hausdorff space if all distinct points in
are pairwise neighbourhood-separable. This condition is the third separation axiom
), which is why Hausdorff spaces are also called
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
points can be separated by disjoint neighbourhoods. Preregular spaces are also called ''
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
For a topological space ''
'', the following are equivalent:
is a Hausdorff space.
* Limits of nets
'' are unique.
* Limits of filters
'' are unique.
* Any singleton set
is equal to the intersection of all closed neighbourhoods
''. (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
Examples and non-examples
Almost all spaces encountered in 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
but is not Hausdorff is the cofinite topology
defined on an infinite set
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.
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 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 T1
spaces in which every convergent sequence has a unique limit.
s and products
of Hausdorff spaces are Hausdorff, but quotient space
s 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
s 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 set
s 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
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
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
and otherwise) to and from Hausdorff spaces.
'' be a continuous function and suppose
is Hausdorff. Then the graph
, is a closed subset of ''
'' be a function and let
be its kernel
regarded as a subspace of ''
'' is continuous and ''
'' is Hausdorff then ''
'' is closed.
'' is an open surjection
'' is closed then ''
'' is Hausdorff.
'' is a continuous, open surjection (i.e. an open quotient map) then ''
'' is Hausdorff if and only if
'' is closed.
'' are continuous maps and ''
'' is Hausdorff then the equalizer
is closed in ''
''. It follows that if ''
'' is Hausdorff and ''
'' and ''
'' agree on a dense
subset of ''
'' then ''
''. In other words, continuous functions into Hausdorff spaces are determined by their values on dense subsets.
'' be a closed
surjection such that ''
'' is compact
for all ''
''. Then if ''
'' is Hausdorff so is ''
'' be a quotient map
'' a compact Hausdorff space. Then the following are equivalent:
'' is Hausdorff.
'' is a closed map
'' is closed.
Preregularity versus regularity
All regular space
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.
The terms "Hausdorff", "separated", and "preregular" can also be applied to such variants on topological spaces as uniform space
s, Cauchy space
s, and convergence space
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
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.
* Hausdorff condition is illustrated by the pun that in Hausdorff spaces any two points can be "housed off" from each other by open sets
[Colin Adams and Robert Franzosa. ''Introduction to Topology: Pure and Applied.'' p. 42]
* In the Mathematics Institute of the University of Bonn
, in which Felix Hausdorff
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.
* , a Hausdorff space ''X'' such that every continuous function has a fixed point.
* Arkhangelskii, A.V., L.S. Pontryagin
, ''General Topology I'', (1990) Springer-Verlag, Berlin. .
; ''Elements of Mathematics: General Topology'', Addison-Wesley (1966).
Category:Properties of topological spaces