Importance of Hausdorffness

This is a survey article related to:Hausdorffness
View other survey articles about 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.

Usefulness of the Hausdorff assumption

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. Further information: Hausdorff implies KC

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 ${\mathbb{R}}^n$, 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. Further information: Hausdorff implies US

Ubiquity of Hausdorffness

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.Further information: Hausdorffness is hereditary

Products of Hausdorff spaces are Hausdorff

This is true whether we use the product topology or the box topology.