This is a survey article describing variations on the following: Hausdorffness
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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 which takes the value at one point, and 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
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.
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.
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.
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.