A topological space (or more generally, a convergence space) is Hausdorff if convergence is unique. The concept can also be defined for locales (see Definition 3 below) and categorified (see Beyond topological spaces below). A Hausdorff space is often called , since this condition came second in the original list of four separation axioms (there are more now) satisfied by metric spaces.
the main separation axioms
|Kolmogorov||given two distinct points, at least one of them has an open neighbourhood not containing the other point||every irreducible closed subset is the closure of at most one point|
|given two distinct points, both have an open neighbourhood not containing the other point||all points are closed|
|Hausdorff||given two distinct points, they have disjoint open neighbourhoods||the diagonal is a closed map|
|and…||all points are closed and…|
|regular Hausdorff||…given a point and a closed subset not containing it, they have disjoint open neighbourhoods||…every neighbourhood of a point contains the closure of an open neighbourhood|
|normal Hausdorff||…given two disjoint closed subsets, they have disjoint open neighbourhoods||…every neighbourhood of a closed set also contains the closure of an open neighbourhood |
… every pair of disjoint closed subsets is separated by an Urysohn function
Hausdorff spaces are a kind of nice topological space; they do not form a particularly nice category of spaces themselves, but many such nice categories consist of only Hausdorff spaces. In fact, Felix Hausdorff's original definition of ‘topological space’ actually required the space to be Hausdorff, hence the name. Certainly homotopy theory (up to weak homotopy equivalence) needs only Hausdorff spaces. It is also common in analysis to assume that all spaces encountered are Hausdorff; if necessary, this can be arranged since every space has a Hausdorff quotient (in fact, the Hausdorff spaces form a reflective subcategory of Top), although usually an easier method is available than this sledgehammer.
There are many equivalent ways of characterizing a space as Hausdorff. The traditional definition is this:
Here is a classically equivalent definition that is more suitable for constructive mathematics:
Given points and of , if every neighbourhood of in meets every neighbourhood of in (which means that is inhabited), then .
This is the mundane way of saying that is closed in .
Another way of saying this, which makes sense also for locales, is the following:
This way of stating the definition generalizes to topos theory and thus to many other contexts; but it is not always a faithful generalization of the classical notion for topological spaces. See Beyond topological spaces below for more.
That is, convergence in a Hausdorff space is unique.
and such that the bijection is constituted by
Here (or more precisely ) is also called the Hausdorff reflection (or Hausdorffication or similar) of .
Moreover, the operation extends to continuous functions
where denotes the equivalence class under of any .
Finally, the comparison map is compatible with this in that the follows squares commute:
There are various ways to see the existence and to construct the Hausdorff reflection. The following is maybe the quickest way to see the existence, even though it leaves the actual construction rather implicit.
Then the set of equivalence classes
First observe that every continuous function into a Hausdorff space factors uniquely via through a continuous function
That this is well defined and continuous follows directly from the definitions.
What remains to be seen is that is indeed a Hausdorff space. Hence assume that . By construction of this means that there exists a Hausdorff space and a surjective continuous function such that . Accordingly, since is Hausdorff, there exist disjoint open neighbourhoods . Moreover, by the previous statement there exists a continuous function with and . Since, by the nature of continuous functions, the pre-images are still disjoint and open, we have found disjoint neighbourhoods of and . Hence is Hausdorff.
Some readers may find the following a more direct way of constructing the Hausdorff reflection:
We start the induction with the trivial equivalence relation:
For a successor ordinal we set
and for a limit ordinal we set
there exists an ordinal such that
for this then is the Hausdorff reflection from prop. 3.
The topology on a compact Hausdorff space is given precisely by the (existent because compact, unique because Hausdorff) limit of each ultrafilter on the space. Accordingly, compact Hausdorff topological spaces are (perhaps surprisingly) described by a (large) algebraic theory. In fact, the category of compact Hausdorff spaces is monadic (over Set); the monad in question maps each set to the set ultrafilters on it. (The results of this paragraph require the ultrafilter theorem, a weak form of the axiom of choice; see ultrafilter monad.)
A compact Hausdorff locale (or space) is necessarily regular; a regular locale (or space) is necessarily Hausdorff. Accordingly, locale theory usually speaks of ‘compact regular’ locales instead of ‘compact Hausdorff’ locales, since the definition of regularity is easier and more natural. Then a version of the previous paragraph works for compact regular locales without the ultrafilter theorem, and indeed constructively over any topos.
Arguably, the desire to make spaces Hausdorff () in analysis is really a desire to make them ; nearly every space that arises in analysis is at least regular, and a regular space must be Hausdorff. Forcing a space to be is like forcing a category to be skeletal; indeed, forcing a preorder to be a partial order is a special case of both (see specialisation topology for how). It may be nice to assume, when working with a particular space, that it is but not to assume, when working with a particular underlying set, that every topology on it is .
Whatever one thinks of that, there is a non- version of Hausdorff space, an space. (The symbol here comes from being a weak version of a regular space; in general a space is precisely both and ). This is also called a preregular space (in HAF) and a reciprocal space (in convergence theory).
Given points and , if every neighbourhood of meets every neighbourhood of , then every neighbourhood of is a neighbourhood of . Equivalently, if any net (or proper filter) converges to both and , then every net (or filter) that converges to also converges to .
There is also a notion of sequentially Hausdorff space:
Whenever a sequence converges to both and , then .
Some forms of predicative mathematics find this concept more useful. Hausdorffness implies sequential Hausdorffness, but the converse is false even for sequential spaces (although it is true for first-countable spaces).
The reader can now easily define a sequentially space.
The most obvious definition for a locale to be Hausdorff is that its diagonal is a closed (and hence proper) inclusion. However, if is a sober space regarded as a locale, this might not coincide with the condition for to be Hausdorff as a space, since the product in the category of locales might not coincide with the product in the category of spaces. But it does coincide if is a locally compact locale, so in that case the two notions of Hausdorff are the same.
This notion of a Hausdorff locale is a special case of that of Hausdorff topos in topos theory. This also includes notions such as a separated scheme etc. The corresponding relative notion (over an arbitrary base topos) is that of separated geometric morphism. For schemes see separated morphism of schemes.
In constructive mathematics, the Hausdorff notion multifurcates further, due to the variety of possible meanings of closed subspace. If we ask the diagonal to be weakly closed, then in the spatial case, this means that it contains all its limit points, giving Definition 2 above. But if we ask the diagonal to be strongly closed, i.e. the complement of an open set, then in the spatial case this means that there is a tight inequality (the exterior? of ) relative to which Definition 1 holds. (We use twice in that definition: in the hypothesis that and in the conclusion that .)
It is natural to call these conditions weakly Hausdorff and strongly Hausdorff, but one should be aware of terminological clashes: in classical mathematics there is a different notion of a weak Hausdorff space, whereas (strong) Hausdorffness for locales has by some authors been called “strongly Hausdorff” only to contrast it with Hausdorffness for spaces.
As a simple example, consider a discrete space regarded as a locale. Since it is locally compact, the locale product coincides with the space product (a theorem that is valid constructively); but nevertheless we have:
In particular, the statement “all discrete locales are localically strongly Hausdorff” is equivalent to excluded middle.
However, non-discrete spaces can constructively be localically strongly Hausdorff without having decidable equality. For instance, any regular space is also regular as a locale, and hence localically strongly Hausdorff. We can also say:
In any topological space , let mean that there exist opens with and and ; then is always an inequality relation. If the spatial product coincides with the locale product (such as if is locally compact), then is localically strongly Hausdorff if and only if is an apartness relation and every open set in is -open (i.e. for any and we have ).
Note that is, as a subset , the exterior of the diagonal in the product topology, mentioned above. If is localically strongly Hausdorff, then must be the open set of which the diagonal is the complementary closed sublocale, since it is the largest open set disjoint from the diagonal.
To say that the diagonal is its complementary closed sublocale implies in particular that for any open set , the open set is the largest open subset of whose intersection with the diagonal is contained in . Specifically, therefore, contains (since is an open subset of whose intersection with the diagonal is ). That is, if and , then either (i.e. ) or (i.e. ). This shows that is -open.
To show that is an apartness, note that for any the set is open, since it is the preimage of under a section of the second projection . Thus, it is -open, which is to say that if then for any either or , which is the missing comparison axiom for to be an apartness.
Conversely, suppose is an apartness and every open set is -open (i.e. the apartness topology refines the given topology on ). Let be an open set; we must show that is the largest open subset of whose intersection with the diagonal is contained in . In other words, suppose is a basic open in and (which is ) is contained in ; we must show . In terms of elements, we assume that if and then , and we must show that if and then .
Assuming and , since and are -open we have either or , and either or . Since we are done if , it suffices to assume and . Therefore, by assumption, and . Since is open in the product topology, we have opens with and and and . But now -openness of and tells us again that either (in which case we are done) or and . In the latter case, (and also ), and hence is also in .
Note that the apartness need not be tight, and in particular need not be spatially Hausdorff. In particular, if might not even be : since localic Hausdorffness is (of course) only a property of the open-set lattice, it only “sees” the sobrification and in particular the quotient (Kolmogorov quotient). However, this is all that can go wrong: if , then by -openness every open set containing must also contain and vice versa, so if is then and is tight.
If the locale product does not coincide with the spatial product, then the “only if” direction of the above proof still works, if we define to be the open part of the locale product given by . A different proof is to recall that by this theorem, an apartness relation is the same as a (strongly) closed equivalence relation on a discrete locale, and the quotient of such an equivalence relation is the -topology. Thus, if is localically strongly Hausdorff, its diagonal is a closed equivalence relation, which yields by pullback a closed equivalence relation on the discrete locale on the same set of points. This is the kernel pair of the canonical surjection , and hence its quotient (the -topology) maps to , i.e. refines the topology of .
A detailed discussion of Hausdorff reflection is in
Comments on the relation to topos theory are for instance in