Hausdorff space

This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces

In the T family (properties of topological spaces related to separation axioms), this is called: T2

This article is about a basic definition in topology.
VIEW: Definitions built on this | Facts about this | Survey articles about this
View a complete list of basic definitions in topology

For survey articles related to this, refer: Category:Survey articles related to Hausdorffness

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

Definition

Symbol-free definition

A topological space is said to be Hausdorff if it satisfies the following equivalent conditions:

Given any two points in the topological space, there are disjoint open sets containing the two points respectively.
The diagonal is closed in the product of the space with itself
Every ultrafilter of subsets converges to at most one point

Definition with symbols

A topological space $X$ is said to be Hausdorff if it satisfies the following equivalent conditions:

Given any two points $x \neq y \in X$ , there exist disjoint open subsets $U ∋ x$ and $V ∋ y$ .
In the product space $X \times X$ , endowed with the product topology, the diagonal, viz., the subset given by ${(x, x) ∣ x \in X}$ is a closed subset
If $S_{α}$ is an ultrafilter of subsets of $X$ , then there is at most one $x \in X$ for which $S_{α} \to x$

Examples

Extreme examples

The empty space is considered a Hausdorff space. For this space, the Hausdorffness condition is vacuously satisfied.
The one-point space is considered 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 considered 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.

Relation with other properties

This property is a pivotal (important) member of its property space. Its variations, opposites, and other properties related to it and defined using it are often studied

Category:Variations of Hausdorffness

Stronger properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Normal space	$T_{1}$ and any two disjoint closed subsets are separated by disjoint open subsets	normal implies Hausdorff	Hausdorff not implies normal	Completely regular space\|FULL LIST, MORE INFO
Collectionwise normal space	$T_{1}$ and any discrete collection of closed subsets is separated by disjoint open subsets	(via normal)	(via normal)	Normal Hausdorff space\|FULL LIST, MORE INFO
Regular space	$T_{1}$ and any point and closed subset not containing it are separated by disjoint open subsets	regular implies Hausdorff	Hausdorff not implies regular	\|FULL LIST, MORE INFO
Completely regular space	$T_{1}$ and any point and disjoint closed subset are separated by a continuous function to $[0, 1]$	(via regular)	(via regular)	\|FULL LIST, MORE INFO
Collectionwise Hausdorff space	any discrete set of points can be separated by disjoint open subsets	collectionwise Hausdorff implies Hausdorff	Hausdorff not implies collectionwise Hausdorff	\|FULL LIST, MORE INFO
Metrizable space	underlying topology of a metric space	metrizable implies Hausdorff	Hausdorff not implies metrizable	Collectionwise Hausdorff space, Completely regular space, Functionally Hausdorff space, Monotonically normal space, Moore space, Normal Hausdorff space, Regular Hausdorff space, Submetrizable space, Tychonoff space\|FULL LIST, MORE INFO
CW-space	underlying topological space of a CW-complex	CW implies Hausdorff	Hausdorff not implies CW	Completely regular space, Normal Hausdorff space, Regular Hausdorff space\|FULL LIST, MORE INFO
Manifold				Metrizable space, Monotonically normal space, Submetrizable space\|FULL LIST, MORE INFO

Weaker properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Locally Hausdorff space	every point is contained in an open subset that's Hausdorff	Hausdorff implies locally Hausdorff	locally Hausdorff not implies Hausdorff	\|FULL LIST, MORE INFO
KC-space	every compact subset is closed	Hausdorff implies KC	KC not implies Hausdorff	Weakly Hausdorff space\|FULL LIST, MORE INFO
US-space	every convergent sequence has a unique limit	Hausdorff implies US	US not implies Hausdorff	KC-space, Weakly Hausdorff space\|FULL LIST, MORE INFO
Sober space	every irreducible subset is the closure of a single point	Hausdorff implies sober	sober not implies Hausdorff	Sober T0 space, Sober T1 space\|FULL LIST, MORE INFO
T1 space	every point is closed	Hausdorff implies T1	T1 not implies Hausdorff	KC-space, Locally Hausdorff space, Sober T1 space, Weakly Hausdorff space\|FULL LIST, MORE INFO
Kolmogorov space (also called $T_{0}$ )				KC-space, Locally Hausdorff space, Sober T0 space\|FULL LIST, MORE INFO

Metaproperties

Metaproperty name	Satisfied?	Proof	Section in this article
product-closed property of topological spaces	Yes	Hausdorffness is product-closed	#Products
box product-closed property of topological spaces	Yes	Hausdorffness is box product-closed	#Box products
subspace-hereditary property of topological spaces	Yes	Hausdorffness is hereditary	#Hereditariness
refining-preserved property of topological spaces	Yes	Hausdorffness is refining-preserved	#Refining

Products

This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products

An arbitrary (finite or infinite) product of Hausdorff spaces, equipped with the product topology, is Hausdorff. For full proof, refer: Hausdorffness is product-closed

Further information: regularity is product-closed, complete regularity is product-closed, T1 is product-closed

Box products

This property of topological spaces is a box product-closed property of topological spaces: it is closed under taking arbitrary box products
View other box product-closed properties of topological spaces

An arbitrary (finite or infinite) product of Hausdorff spaces, equipped with the box topology, is regular. For full proof, refer: Hausdorffness is box product-closed

Further information: regularity is box product-closed, Urysohn is box product-closed

Hereditariness

This property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.
View other subspace-hereditary properties of topological spaces

Any subspace of a Hausdorff space is Hausdorff. For full proof, refer: Hausdorffness is hereditary

Refining

This property of topological spaces is preserved under refining, viz, if a set with a given topology has the property, the same set with a finer topology also has the property
View all refining-preserved properties of topological spaces OR View all coarsening-preserved properties of topological spaces

Moving to a finer topology increases the number of possible open sets to choose from, and hence, preserves the property of Hausdorffness. For full proof, refer: Hausdorffness is refining-preserved

References

Textbook references

Topology (2nd edition) by James R. Munkres^{More info}, Page 98, Chapter 2, Section 17 (formal definition)
Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 26 (formal definition)

External links

Definition links

Wikipedia page