Compact Hausdorff space: Difference between revisions
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| 1 || Compact and Hausdorff || it is [[compact space|compact]] (i.e., every [[open cover]] has a finite subcover) and [[Hausdorff space|Hausdorff]] (i.e., any two distinct points can be separated by disjoint open subsets) || {{fillin}} | | 1 || Compact and Hausdorff || it is [[compact space|compact]] (i.e., every [[open cover]] has a finite subcover) and [[Hausdorff space|Hausdorff]] (i.e., any two distinct points can be separated by disjoint open subsets) || {{fillin}} | ||
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| 2 || Closed iff compact on subsets || a subset is [[defining ingredient::closed subset|closed]] iff it is [[defining ingredient::compact space|compact]] in the [[defining ingredient::subspace topology]] || for any subset <math>A</math> of <math>X</math>, <math>A</math> is [[closed subset|closed]] in <math>X</math> if and only if it is a [[compact space]] in the [[subspace topology]] | | 2 || Closed iff compact on subsets || a subset is [[defining ingredient::closed subset|closed]] iff it is [[defining ingredient::compact space|compact]] in the [[defining ingredient::subspace topology]] || for any subset <math>A</math> of <math>X</math>, <math>A</math> is [[closed subset|closed]] in <math>X</math> if and only if it is a [[compact space]] in the [[subspace topology]]. | ||
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| 3 || Ultrafilter formulation || every ultrafilter converges to a unique point. || for any ultrafilter <math>S</math> of <math>X</math>, there is a unique point <math>x \in X</math> such that <math>S \to x</math>. | | 3 || Ultrafilter formulation || every ultrafilter converges to a unique point. || for any ultrafilter <math>S</math> of <math>X</math>, there is a unique point <math>x \in X</math> such that <math>S \to x</math>. | ||
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| 4 || Closed subspace of Tychonoff cube || it is homeomorphic to a closed subspace of a Tychonoff cube. || there exists a power <math>\lambda</math> such that <math>[0,1]^\lambda</math> has a closed subspace <math>C</math> homeomorphic to <math>X</math>. | |||
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==Relation with other properties== | ==Relation with other properties== | ||
===Stronger properties=== | ===Stronger properties=== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||
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| [[Stronger than::compact T1 space]] || || || || | | [[Stronger than::compact space]] || every open cover has a finite subcover || || || {{intermediate notions short|compact space|compact Hausdorff space}} | ||
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| [[Stronger than::compact T1 space]] || || || [[cofinite topology]] on an infinite set is compact and T1 but not Hausdorff || {{intermediate notions short|compact T1 space|compact Hausdorff space}} | |||
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| [[Stronger than::Baire space]] || Any countable intersection of dense open subsets is dense || || || {{intermediate notions short|Baire space|compact Hausdorff space}} | | [[Stronger than::Baire space]] || Any countable intersection of dense open subsets is dense || || || {{intermediate notions short|Baire space|compact Hausdorff space}} | ||
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| [[Stronger than::paracompact Hausdorff space]] || [[paracompact space|paracompact]] and [[Hausdorff space|Hausdorff]] || || || | | [[Stronger than::paracompact Hausdorff space]] || [[paracompact space|paracompact]] and [[Hausdorff space|Hausdorff]] || || || | ||
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| [[Stronger than::binormal space]] || product with the [[unit interval]] is a [[normal space]] || ([[paracompact Hausdorff implies binormal|via paracompact Hausdorff]]) || || {{intermediate notions short|binormal space|compact Hausdorff space}} | |||
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| [[Stronger than::normal space]] || [[Hausdorff space|Hausdorff]] and any two disjoint closed subsets can be separated by disjoint open subsets. || [[compact Hausdorff implies normal]] || || {{intermediate notions short|normal space|compact Hausdorff space}} | | [[Stronger than::normal space]] || [[Hausdorff space|Hausdorff]] and any two disjoint closed subsets can be separated by disjoint open subsets. || [[compact Hausdorff implies normal]] || || {{intermediate notions short|normal space|compact Hausdorff space}} | ||
Latest revision as of 22:41, 29 April 2022
Definition
Equivalent definitions in tabular format
| No. | Shorthand | A topological space is said to be compact Hausdorff if ... | A topological space is said to be compact Hausdorff if ... |
|---|---|---|---|
| 1 | Compact and Hausdorff | it is compact (i.e., every open cover has a finite subcover) and Hausdorff (i.e., any two distinct points can be separated by disjoint open subsets) | Fill this in later |
| 2 | Closed iff compact on subsets | a subset is closed iff it is compact in the subspace topology | for any subset of , is closed in if and only if it is a compact space in the subspace topology. |
| 3 | Ultrafilter formulation | every ultrafilter converges to a unique point. | for any ultrafilter of , there is a unique point such that . |
| 4 | Closed subspace of Tychonoff cube | it is homeomorphic to a closed subspace of a Tychonoff cube. | there exists a power such that has a closed subspace homeomorphic to . |
![{\displaystyle [0,1]^{\lambda }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/336ca507818511369e34330369173300172ba4c9)
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
This article describes a property of topological spaces obtained as a conjunction of the following two properties: compactness and Hausdorffness
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| compact metrizable space | |FULL LIST, MORE INFO | |||
| compact polyhedron | |FULL LIST, MORE INFO | |||
| compact manifold | |FULL LIST, MORE INFO |
Weaker properties
![{\displaystyle [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d)
Metaproperties
Products
This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products
An arbitrary product of compact Hausdorff spaces is compact Hausdorff. This follows from independent statements to that effect for compactness, and for Hausdorffness.
Weak hereditariness
This property of topological spaces is weakly hereditary or closed subspace-closed; in other words, any closed subset (equipped with the subspace topology) of a space with the property, also has the property.
View all weakly hereditary properties of topological spaces | View all subspace-hereditary properties of topological spaces
A closed subset of a compact Hausdorff space is compact Hausdorff. In fact, a subset is compact Hausdorff iff it is closed:
- Every subspace is anyway Hausdorff
- Since the whole space is compact, any closed subset is compact
- Since the whole space is Hausdorff, any compact subset is closed
Effect of property operators
The subspace operator
Applying the subspace operator to this property gives: completely regular space
A topological space can be embedded in a compact Hausdorff space iff it is completely regular. Necessity follows from ths fact that compact Hausdorff spaces are completely regular, and any subspace of a completely regular space is completely regular. Sufficiency follows from an explicit construction, such as the Stone-Cech compactification.