Hausdorff space

In topology and related branches of mathematics, Hausdorff spaces and preregular spaces are kinds of topological spaces. The conditions imposed are the most significant separation axioms.

Definitions

Suppose that X is a topological space.

X is a Hausdorff space, or T₂ 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.

The points x and y, here represented by dots on opposite sides of the picture, are separated by their respective neighbourhoods U and V, here represented by disjoint open disks with the original dots at their centres.

X is a preregular space, or R₁ space, iff, given any topologically distinguishable points x and y, x and y can be separated by neighbourhoods.

Alternative characterisations

Limits of sequences, nets, and filters (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 T₀. 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 T₁, meaning that all singletons are closed. Similarly, preregular spaces are R₀.

Subspaces 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

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 (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.

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 (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 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).

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. Atiyah used to draw house-shaped sets, on the blackboard. (In an old-fashioned British accent, off could be orf, phonetically, which all helps.)