Importance of Hausdorffness

Introduction
Hausdorffness is one of the most important properties of topological spaces; in fact, a number of texts assume Hausdorffness as one of the axioms that any topological space must satisfy. In this article, we see a number of ways in which Hausdorffness is a good property to have, why we can remain within the world of Hausdorff spaces for a number of purposes, and why it is sometimes important to transcend the world of Hausdorff spaces.

The ability to separate points
Hausdorff spaces are spaces where the intuition that any two points are far off, is concretely realized: given any two points, there are disjoint open sets containing both of them. A number of topological spaces encountered are Hausdorff.

Compact subsets are closed
One of the most frequent uses of the Hausdorffness assumption is that compact subsets of a Hausdorff space are closed. The power of this is that knowing something about a subset as an abstract topological space, we can deduce that it is embedded as a closed subset.

Hausdorffness is not a necessary condition for compact subsets to be closed; in general, a topological space for which compact subsets are closed is termed a KC-space. However, Hausdorffness is practically the most common way of ensuring that compact subsets are closed.

The fact that compact subsets are closed is used in a number of ways; most frequently when using a gluing lemma with local data. For instance, if $$x$$ is a Euclidean point in a Hausdorff space, and $$U$$ is a neighbourhood of $$x$$ homeomorphic to $$\R^n$$, then the image of the closed disc in $$U$$ is a closed subset of the whole space. This idea is used in arguments involving manifolds and CW-spaces; for instance the proof that any connected manifold is homogeneous.

Sequences have unique limits
In a Hausdorff space, it makes sense to talk of the limit of a sequence. In other words, the same sequence of points cannot have two different limits. This is essentially because any two distinct points are separated by disjoint open subsets.

Again, Hausdorffness is not a necessary condition for sequences to have unique limits. A topological space in which every sequence of points has a unique limit is termed a US-space. However, Hausdorffness is practically the most common way of ensuring that sequences of points have unique limits.

Jacking up from Hausdorffness
Often, assuming Hausdorffness allows us to prove much stronger separation assumptions, such as normality, or even metrizability. The key idea is to combine Hausdorffness with other local or global data given. For instance, Hausdorffness combines well with compactness-type assumptions:


 * Any compact Hausdorff space is normal: The proof of this involves first choosing a pair of disjoint open subsets for each point, and then passing to a finite subcover using compactness.
 * More generally, any paracompact Hausdorff space is normal: The proof is almost identical.

Other examples of where Hausdorffness gives us base leverage are CW-spaces and manifolds. The Hausdorffness assumption, along with second-countability, allows us to prove that manifolds can be embedded inside Euclidean space, which shows that they are metrizable, normal, and many other things.

Uniform spaces
Topological spaces which arise from uniform spaces, such as metrizable spaces and the underlying spaces of topological groups, are Hausdorff. The idea is to use the fact that an open set containing one point that does not contain the other can be halved to get two open sets, one containing each point. (In fact, the underlying topological space of a uniform space must be completely regular).

CW-spaces
CW-complexes are often considered the building blocks of algebraic topology, because any topological space is weakly homotopy equivalent to a CW-complex. CW-complexes are the approximations to topological spaces which preserve the homology and homotopy. It turns out that CW-spaces are Hausdorff (in fact, they are normal, and much more).

Manifolds
Manifolds are Hausdorff -- this is part of the definition. If we consider locally Euclidean spaces that are not Hausdorff, a number of pathologies can occur; however, adding the Hausdorff assumption, along with the assumption of being second-countable, allows us to deduce powerful results like: any manifold can be embedded as a closed subset of Euclidean space. Thus, the condition of Hausdorffness gives us enough leverage to prove that manifolds are metrizable, normal, and many other things.

Closed nature of Hausdorffness
If we start with Hausdorff spaces, and remain in the world of Hausdorff spaces, it is in general difficult to escape this world.

Subspaces of Hausdorff spaces are Hausdorff
If our universe is a Hausdorff space, then any subset of that universe is also Hausdorff.

Products of Hausdorff spaces are Hausdorff
This is true whether we use the product topology or the box topology.

Not a local property
If we build a space whose local models are Hausdorff, the whole space need not be Hausdorff. This is because Hausdorffness is a global property. Thus, it is possible to build locally Hausdorff spaces which are not Hausdorff. More worrisome is the fact that even if the space we build is Hausdorff, there is in general no local proof of the fact; we need a global argument.

An example is locally Euclidean spaces. There are locally Euclidean spaces which are not Hausdorff, for instance, the line with two origins. For those locally Euclidean spaces which are Hausdorff, a separate proof of Hausdorffness is usually required.

Not preserved on taking quotients
A quotient of a Hausdorff space under an equivalence relation is not necessarily Hausdorff, even if we assume good things about the equivalence relation. Even in the cases that the quotient happens to be Hausdorff, we usually need to prove the fact by hand. This is closely related to Hausdorffness not being a local property, because the preservation of local properties under quotient maps is easy to ensure.

A number of non-Hausdorff (in fact, even non-T1) quotients arise when we consider group actions on a topological space. Such actions occur in algebraic geometry; for instance, the action of the group of multiplicative reals on $$\R^n$$, gives the point zero, along with the projective space of dimension $$n-1$$. The projective space of dimension $$n-1$$ is a manifold under the subspace topology; however, the point zero is a generic point: its closure is the whole space.

In fact, a number of constructions in algebraic geometry, such as the spectrum of a ring, yield non-Hausdorff spaces. Often, we can find a large Hausdorff subspace, and concentrate on that subspace (in the above example, the subspace is the complement of the zero point).

Not satisfied for étale spaces
In a number of constructions, the étale spaces associated with sheaves are not Hausdorff. For instance, given two reasonably nice topological spaces $$X$$ and $$Y$$, the étale space of continuous functions from $$X$$ to $$Y$$ is usually not Hausdorff. The problem, in general, is that we may have two functions whose germ at a point is not equal, but whose germs are equal at points arbitrarily close to it.