Hausdorff space

Please also read the Topospaces Convention page: Convention:Hausdorffness assumption

Equivalent definitions in tabular format
A topological space is said to be Hausdorff if it satisfies the following equivalent conditions:

Extreme examples

 * The empty space is a Hausdorff space. For this space, the Hausdorffness condition is vacuously satisfied.
 * The one-point space is a Hausdorff space. For this space, the Hausdorffness condition is vacuously satisfied.
 * Any discrete space (i.e., a topological space with the discrete topology) is a Hausdorff space.

Typical examples

 * Euclidean space, and more generally, any manifold, closed subset of Euclidean space, and any subset of Euclidean space is Hausdorff.
 * Any metric space is Hausdorff in the induced topology, i.e., any metrizable space is Hausdorff.

Non-examples

 * The spectrum of a commutative unital ring is generally not Hausdorff under the Zariski topology.
 * The etale space of continuous functions, and more general etale spaces, are usually not Hausdorff.

Opposite properties

 * Irreducible space: See irreducible and Hausdorff implies one-point space
 * Ultraconnected space: See ultraconnected and T1 implies one-point space

Textbook references

 * , Page 98, Chapter 2, Section 17 (formal definition)
 * , Page 26 (formal definition)