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:
Category:Variations of Hausdorffness
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
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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.