# Varying Hausdorffness

This is a survey article describing variations on the following: Hausdorffness
View other variational survey articles | View other survey articles about Hausdorffness

Hausdorffness is one of the more pivotal properties of topological spaces; some people go so far as to require it in the definition of topological space. A number of variations of Hausdorffness have been considered, and carefully studied, by topologists. A list of some of the properties obtained by varying Hausdorffness is at:

This article surveys some of the common variations of Hausdorffness, trying to organize them into themes and streams. There are three basic ideas behind variation:

• Emulate the strengths
• Remedy the weaknesses
• Weaken or remove the strengths

## The extent of separation

### How well two points are separated

Further information: Urysohn space

Hausdorffness allows us to separate points by disjoint open subsets. In fancy language, a space is Hausdorff if any two points are separated subsets. However, there are situations where we want a stronger level of separation.

• Urysohn space is a topological space where, given any two points, there is a continuous function to $[0,1]$ which takes the value $0$ at one point, and $1$ at the other. Separation by a continuous function is a much stronger assumption, and allows us to deduce much more. For instance, a connected Urysohn space with more than one point must be uncountable. Further information: connected Urysohn implies uncountable

### Separating more than two points

Further information: Collectionwise Hausdorff space

A useful variant of Hausdorffness is the property of being a collectionwise Hausdorff space: a T1 space in which, given any discrete closed subset, there exist pairwise disjoint open sets containing each point.

### Separating bigger subsets

The attempt to separate subsets that are bigger in size than points leads to the other separation axioms, such as regularity and normality.

## Consequences of Hausdorffness

The strength of Hausdorffness lies in a number of facts we can prove for Hausdorff spaces, such as:

• Any compact subset of a Hausdorff space is closed
• Every convergent sequence in a Hausdorff space has a unique limit

It is natural to ask what we can say about topological spaces for which we simply assume these properties.

### KC-spaces

Further information: KC-space, Hausdorff implies KC

A KC-space is a topological space in which every compact subset is closed. Compact subsets being closed is one of the most useful implications of Hausdorffness; for instance, it is the key to proving that connected manifolds are homogeneous and that the inclusion of a point in a manifold is a cofibration.

Thus, many of the good things we prove about Hausdorff spaces, continue to be valid for KC-spaces. However, non-Hausdorff KC-spaces arise very rarely in most applications.

### US-spaces

Further information: US-space, Hausdorff implies US

A US-space is a topological space in which every sequence has at most one limit (or equivalently, every convergent sequence has a unique limit). Hausdorff spaces are US-spaces, because any two points are far-off. But there exist US-spaces which are not Hausdorff, and again, many proofs which use the uniqueness of limit for Hausdorff spaces, can generalize to US-spaces. Non-Hausdorff US-spaces arise very rarely in most applications.

### Sober spaces

Further information: Sober space

A sober space is a topological space in which every irreducible closed subset is the closure of a one-point set. For Hausdorff spaces, the only irreducible closed subsets are the one-point sets, so Hausdorff spaces are clearly closed. Non-Hausdorff sober spaces arise as the spectrum of a ring.