Encyclopedia
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. Of the many
separation axioms that can be imposed on a topological space, the Hausdorff condition is the most frequently used and discussed .
Hausdorff spaces are named for
Felix Hausdorff, one of the founders of topology. In fact, Hausdorff's original definition of a topological space included the Hausdorff condition as an axiom.
Definitions
Suppose that
X is a topological space. Let
x and
y be points in
X. 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 .
X is a
Hausdorff space if any two distinct points of
X can be separated by neighborhoods. 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. 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 . A topological space is preregular if and only if its Kolmogorov quotient is Hausdorff.
Examples and counterexamples
Almost all spaces encountered in analysis are Hausdorff; most importantly, the 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 T
1 but is not Hausdorff is the cofinite topology.
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.
Properties
One of the nicest properties of Hausdorff spaces is that limits of sequences,
nets, and filters are unique whenever they exist. In fact, a topological space is Hausdorff if and only if every net has at most one limit. Similarly, a space is preregular if all of the limits of a given net are topologically indistinguishable.
A useful alternative characterization of Hausdorff spaces is the following. A topological space
X is Hausdorff if and only if the diagonal ? = is closed as a subset of the
product space X ×
X.
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 T
1, meaning that all singletons are closed. Similarly, preregular spaces are R
0.
Another nice property of Hausdorff spaces is that compact sets are always closed. This may fail for spaces which are non-Hausdorff .
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. 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 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. 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 to and from Hausdorff spaces.
Let
f :
X ?
Y be a function and let be its kernel regarded as a subspace of
X ×
X.
- If f is continuous and Y is Hausdorff then ker is closed.
- If f is an open surjection
- injective function [i] ...
and ker is closed then Y is Hausdorff. - If f is a continuous, open surjection then Y is Hausdorff if and only if ker is closed.
If
f,g :
X ?
Y are continuous maps and
Y is Hausdorff then the equalizer 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 ?
Y be a closed surjection such that
f−1 is compact for all
y ?
Y. Then if
X is Hausdorff so is
Y.
Let
f :
X ?
Y be a
quotient map with
X a compact Hausdorff space. Then the following are equivalent
- Y is Hausdorff
- f is a closed map
- ker 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 don't also apply to nonregular Hausdorff spaces.
There are many situations where another condition of topological spaces will imply regularity if preregularity is satisfied.
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 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.
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 are unique or unique up to topological indistinguishability .
As it turns out, uniform spaces, and more generally Cauchy spaces, are always preregular, so the Hausdorff condition in these cases reduces to the T
0 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 .
Joke
There is a mathematicians' joke that serves as a reminder of the meaning of this term:
In a Hausdorff space, points can be "housed off" from one another.
Michael Atiyah used to draw house-shaped sets on the blackboard.
