# Hausdorff space

In topology and related branches of mathematics,**Hausdorff spaces**and

**preregular spaces**are particularly nice kinds of topological spaces. These conditions are examples of separation axioms.

Hausdorff spaces are named after Felix Hausdorff, who helped originate general topology. In fact, Hausdorff's original definition of topological space required all topological spaces to be Hausdorff (a requirement that is not made today).

Table of contents |

2 Examples and nonexamples 3 Properties 4 Variations 5 Joke |

## Definitions

Suppose that *X* is a topological space.

*X* is a *Hausdorff space*, or *T _{2} space*, or

*separated space*, iff, given any distinct points

*x*and

*y*, there are a neighbourhood

*U*of

*x*and a neighbourhood

*V*of

*y*that are disjoint. In fancier terms, this condition says that

*x*and

*y*can be

*separated by neighbourhoods*.

*X* is a *preregular space*, or *R _{1} space*, iff, given any topologically distinguishable points

*x*and

*y*,

*x*and

*y*can be separated by neighbourhoods.

### Alternative characterisations

Limitss of sequences, netss, and filterss (when they exist) are unique in Hausdorff spaces. In fact, a topological space is Hausdorff if and only if every net (or filter) has at most one limit. Similarly, a space is preregular iff all of the limits of a given net (or filter) are topologically indistinguishable.

A topological space *X* is Hausdorff if and only if the diagonal {(*x*,*x*) : *x* in *X*} is a closed set in *X* × *X*, the Cartesian product of *X* with itself.

A topological space is Hausdorff if and only if it is both preregular and T_{0}.
Conversely, a topological space is preregular if and only if its Kolmogorov quotient is Hausdorff.

## Examples and nonexamples

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. 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

Hausdorff spaces are T_{1}, meaning that all singletonss are closed.
Similarly, preregular spaces are R_{0}.

Subspacess and products of Hausdorff spaces are Hausdorff, but quotient spaces of Hausdorff spaces need not be Hausdorff.
In fact, a quotient space of a Hausdorff space *X* is itself Hausdorff if and only if the kernel of the quotient map is closed as a subset of the Cartesian product *X* × *X*.

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.

### Preregularity versus regularity

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 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.

## Variations

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 iff every Cauchy net has at *least* one limit, while a space is Hausdorff iff every Cauchy net has at *most* one limit (since only Cauchy nets can have limits in the first place).