Compact Hausdorff space: Difference between revisions

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 $X$ 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 $A$ of $X$ , $A$ is closed in $X$ 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 $S$ of $X$ , there is a unique point $x \in X$ such that $S \to x$ .
4	Closed subspace of Tychonoff cube	it is homeomorphic to a closed subspace of a Tychonoff cube.	there exists a power $λ$ such that $[0, 1]^{λ}$ has a closed subspace $C$ homeomorphic to $X$ .

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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
compact space	every open cover has a finite subcover			Compact T1 space\|FULL LIST, MORE INFO
compact T1 space			cofinite topology on an infinite set is compact and T1 but not Hausdorff	\|FULL LIST, MORE INFO
Baire space	Any countable intersection of dense open subsets is dense			\|FULL LIST, MORE INFO
locally compact Hausdorff space	locally compact and Hausdorff			\|FULL LIST, MORE INFO
paracompact Hausdorff space	paracompact and Hausdorff
binormal space	product with the unit interval is a normal space	(via paracompact Hausdorff)		Paracompact Hausdorff space\|FULL LIST, MORE INFO
normal space	Hausdorff and any two disjoint closed subsets can be separated by disjoint open subsets.	compact Hausdorff implies normal		Binormal space, Paracompact Hausdorff space\|FULL LIST, MORE INFO
completely regular space	Hausdorff and there is a continuous function to $[0, 1]$ separating any point and closed subset disjoint from it.	(via normal)		Normal Hausdorff space, Paracompact Hausdorff space, Tychonoff space\|FULL LIST, MORE INFO

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.

@@ Line 1: / Line 1: @@
-{{topospace property}}
+==Definition==
+===Equivalent definitions in tabular format===
-==Definition==
+{| class="sortable" border="1"
+! No. !! Shorthand !! A [[topological space]] is said to be compact Hausdorff if ... !! A [[topological space]] <math>X</math> is said to be compact Hausdorff if ...
+|-
+| 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}}
+|-
+| 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]].
+|-
+| 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>.
+|-
+| 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>.
+|}
-A [[topological space]] is termed '''compact Hausdorff''' if it satisfies the following equivalent conditions:
+{{basicdef}}
-* It is [[compact space|compact]] and [[Hausdorff space|Hausdorff]]
+{{topospace property conjunction|compactness|Hausdorffness}}
-* A subset is [[closed subset|closed]] iff it is [[compact space|compact]] in the [[subspace topology]]
 ==Relation with other properties==
 ===Stronger properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::compact metrizable space]] || || || ||{{intermediate notions short|compact Hausdorff space|compact metrizable space}}
+|-
+| [[Weaker than::compact polyhedron]] || || || || {{intermediate notions short|compact Hausdorff space|compact polyhedron}}
+|-
+| [[Weaker than::compact manifold]] || || || || {{intermediate notions short|compact Hausdorff space|compact manifold}}
+|}
+===Weaker properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::compact space]] || every open cover has a finite subcover || || || {{intermediate notions short|compact space|compact Hausdorff space}}
+|-
+| [[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}}
+|-
+| [[Stronger than::Baire space]] || Any countable intersection of dense open subsets is dense || || || {{intermediate notions short|Baire space|compact Hausdorff space}}
+|-
+| [[Stronger than::locally compact Hausdorff space]] || [[locally compact space|locally compact]] and [[Hausdorff space|Hausdorff]] || || || {{intermediate notions short|locally compact Hausdorff space|compact Hausdorff space}}
+|-
+| [[Stronger than::paracompact Hausdorff space]] || [[paracompact space|paracompact]] and [[Hausdorff space|Hausdorff]] || || ||
+|-
+| [[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}}
+|-
+| [[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::completely regular space]] || [[Hausdorff space|Hausdorff]] and there is a continuous function to <math>[0,1]</math> separating any point and closed subset disjoint from it. || (via normal) || || {{intermediate notions short|completely regular space|compact Hausdorff space}}
+|}
+==Metaproperties==
+{{DP-closed}}
+An arbitrary product of compact Hausdorff spaces is compact Hausdorff. This follows from independent statements to that effect for compactness, and for Hausdorffness.
-* [[Compact metrizable space]]
+{{closed subspace-closed}}
-===Weaker properties===
+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==
+{{applyingoperatorgives|subspace operator|completely regular space}}
-* [[Baire space]]
+A topological space can be embedded in a compact Hausdorff space iff it is [[completely regular space|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]].
-* [[Paracompact Hausdorff space]]
-* [[Normal space]]