Hausdorff space

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

Definition

Equivalent definitions in tabular format

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

No.	Shorthand	A topological space is termed Hausdorff if ...	A topological space $X$ is termed Hausdorff if ...
1	Separation axiom	given any two distinct points in the topological space, there are disjoint open sets containing the two points respectively.	given any two points $x\neq y\in X$, there exist open subsets $U\ni x$ and $V\ni y$ such that $U\cap V$ is empty
2	Diagonal in square	the diagonal is closed in the product of the space with itself	in the product space $X\times X$, endowed with the product topology, the diagonal, viz., the subset given by $\{(x,x)\mid x\in X\}$ is a closed subset
3	Ultrafilter convergence	every ultrafilter of subsets converges to at most one point	if $S_{\alpha }$ is an ultrafilter of subsets of $X$, then there is at most one $x\in X$ for which $S_{\alpha }\to x$.
4	Separation axiom for finite subsets	given any finite collection of distinct points the topological space, there are open subsets containing each of the points in that finite subset that have pairwise trivial intersection.	for any finite collection $x_{1},x_{2},\dots ,x_{n}$ of distinct points in $X$, there exists a finite collection $U_{1},U_{2},\dots ,U_{n}$ of open subsets of $X$ such that $U_{i}\ni x_{i}$ for all $i$ and$U_{i}\cap U_{j}$ is empty for $i\neq j$.
5	Separation axiom using basis open subsets	(choose a basis of open subsets for the topological space) given any two distinct points in the topological space, there are disjoint open sets from the basis containing the two points respectively.	(choose a basis of open subsets for the topological space) given any two points $x\neq y\in X$, there exist open subsets $U\ni x$ and $V\ni y$ such that both $U$ and $V$ are in the basis and $U\cap V$ is empty
6	Separation axiom for finite subsets using basis open subsets	(choose a basis of open subsets for the topological space) given any finite collection of distinct points the topological space, there are open subsets (all from the basis) containing each of the points in that finite subset that have pairwise trivial intersection.	for any finite collection $x_{1},x_{2},\dots ,x_{n}$ of distinct points in $X$, there exists a finite collection $U_{1},U_{2},\dots ,U_{n}$ of open subsets of $X$ (all from the basis) such that $U_{i}\ni x_{i}$ for all $i$ and$U_{i}\cap U_{j}$ is empty for $i\neq j$.
7	Separation axiom for two compact subsets	any two disjoint compact subsets can be separated by disjoint open subsets	given any compact subsets $A_{1},A_{2}$ of $X$ (i.e., $A_{1},A_{2}$ are both compact spaces in the subspace topology), there exist disjoint open subsets $U_{1},U_{2}$ such that $U_{1}\supseteq A_{1}$ and $U_{2}\supseteq A_{2}$.
8	Separation axiom for finitely many compact subsets	any finite collection of pairwise disjoint compact subsets can be separated by a finite collection of pairwise disjoint open subsets	given a finite collection $A_{1},A_{2},\dots ,A_{n}$ of pairwise disjoint compact subsets of $X$, there exist pairwise disjoint open subsets $U_{1},U_{2},\dots ,U_{n}$ of $X$ such that $A_{i}\subseteq U_{i}$ for each $i$.
9	Separation axiom between point and compact subset	given any point and any compact subset not contianing it, the point and the compact subset can be separated by disjoint open subsets.	for any $x\in X$ and any $A\subseteq X$ such that $A$ is compact and $x\notin A$, there exist disjoint open subsets $U,V$ of $X$ such that $U\ni x,V\supseteq A$.
10	preregular and Kolmogorov	it is both a preregular space and a Kolmogorov space (i.e., a $T_{0}$ space)

Equivalence of definitions

Further information: equivalence of definitions of Hausdorff space

Examples

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.

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

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
product-closed property of topological spaces	Yes	Hausdorffness is product-closed	If $X_{i},i\in I$ is a (finite or infinite) collection of Hausdorff topological spaces, the product of all the $X_{i}$s, equipped with the product topology, is also Hausdorff.
box product-closed property of topological spaces	Yes	Hausdorffness is box product-closed	If $X_{i},i\in I$ is a (finite or infinite) collection of Hausdorff topological spaces, the product of all the $X_{i}$s, equipped with the box topology, is also Hausdorff.
subspace-hereditary property of topological spaces	Yes	Hausdorffness is hereditary	Suppose $X$ is a Hausdorff space and $A$ is a subset of $X$. Under the subspace topology, $A$ is also Hausdorff.
refining-preserved property of topological spaces	Yes	Hausdorffness is refining-preserved	Suppose $\tau _{1}$ and $\tau _{2}$ are two topologies on a set $X$, such that $\tau _{1}\subseteq \tau _{2}$, i.e., every subset of $X$ open with respect to $\tau _{1}$ is also open with respect to $\tau _{2}$. Then, if $X$ is Hausdorff with respect to $\tau _{1}$, it is also Hausdorff with respect to $\tau _{2}$.
local property of topological spaces	No	Hausdorffness is not local	It is possible to have a topological space $X$ such that, for every $x\in X$, there exists an open subset $U\ni x$ that is Hausdorff but $X$ is not itself Hausdorff. Spaces with this property are called locally Hausdorff spaces.
closure-local property of topological spaces	Yes	Hausdorffness is closure-local	If $X$ is a topological space such that, for every $x\in X$, there exists an open subset $U\ni x$ such that the closure ${\overline {U}}$ is Hausdorff, then $X$ is Hausdorff.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal Hausdorff space	$T_{1}$ and any two disjoint closed subsets are separated by disjoint open subsets		Hausdorff not implies normal	Functionally Hausdorff space, Regular Hausdorff space, Tychonoff space, Urysohn space|FULL LIST, MORE INFO
collectionwise normal Hausdorff space	$T_{1}$ and collectionwise normal: any discrete collection of closed subsets is separated by disjoint open subsets	(via normal)	(via normal)	|FULL LIST, MORE INFO
regular Hausdorff space	$T_{1}$ and any point and closed subset not containing it are separated by disjoint open subsets		Hausdorff not implies regular	Urysohn space|FULL LIST, MORE INFO
Tychonoff space	$T_{1}$ and completely regular: any point and disjoint closed subset are separated by a continuous function to $[0,1]$	(via regular)	(via regular)	Functionally Hausdorff space, Regular Hausdorff space, Urysohn space|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
functionally Hausdorff space	any two distinct points can be separated by a continuous function to the reals			Urysohn space|FULL LIST, MORE INFO
Urysohn space	any two distinct points can be separated by closure-disjoint open subsets			|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
submetrizable space	either metrizable or can be made metrizable by passage to a coarser topology			Functionally Hausdorff space|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
preregular space	any two topologically distinguishable points can be separated by pairwise disjoint open subsets	direct		|FULL LIST, MORE INFO
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
weakly Hausdorff space	any subset of it that is the image of a compact Hausdorff space under a continuous map is closed in it			|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
sober T1 space	sober and T1	(via both components of conjunction)		|FULL LIST, MORE INFO
Kolmogorov space (also called $T_{0}$)	any two distinct points are topologically distinguishable			KC-space, Locally Hausdorff space, Sober T0 space|FULL LIST, MORE INFO
sober T0 space	sober and Kolmogorov			Sober T1 space|FULL LIST, MORE INFO

Opposite properties

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

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)