Hausdorff space
From Topospaces
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 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 , there exist open subsets and such that is empty |
2 | Diagonal in square | the diagonal is closed in the product of the space with itself | in the product space , endowed with the product topology, the diagonal, viz., the subset given by is a closed subset |
3 | Ultrafilter convergence | every ultrafilter of subsets converges to at most one point | if is an ultrafilter of subsets of , then there is at most one for which . |
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 of distinct points in , there exists a finite collection of open subsets of such that for all and is empty for . |
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 , there exist open subsets and such that both and are in the basis and 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 of distinct points in , there exists a finite collection of open subsets of (all from the basis) such that for all and is empty for . |
7 | Separation axiom for two compact subsets | any two disjoint compact subsets can be separated by disjoint open subsets | given any compact subsets of (i.e., are both compact spaces in the subspace topology), there exist disjoint open subsets such that and . |
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 of pairwise disjoint compact subsets of , there exist pairwise disjoint open subsets of such that for each . |
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 and any such that is compact and , there exist disjoint open subsets of such that . |
10 | preregular and Kolmogorov | it is both a preregular space and a Kolmogorov space (i.e., a 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 is a (finite or infinite) collection of Hausdorff topological spaces, the product of all the s, equipped with the product topology, is also Hausdorff. |
box product-closed property of topological spaces | Yes | Hausdorffness is box product-closed | If is a (finite or infinite) collection of Hausdorff topological spaces, the product of all the s, equipped with the box topology, is also Hausdorff. |
subspace-hereditary property of topological spaces | Yes | Hausdorffness is hereditary | Suppose is a Hausdorff space and is a subset of . Under the subspace topology, is also Hausdorff. |
refining-preserved property of topological spaces | Yes | Hausdorffness is refining-preserved | Suppose and are two topologies on a set , such that , i.e., every subset of open with respect to is also open with respect to . Then, if is Hausdorff with respect to , it is also Hausdorff with respect to . |
local property of topological spaces | No | Hausdorffness is not local | It is possible to have a topological space such that, for every , there exists an open subset that is Hausdorff but 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 is a topological space such that, for every , there exists an open subset such that the closure is Hausdorff, then is Hausdorff. |
Relation with other properties
Stronger properties
Weaker properties
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)