Compact space: Difference between revisions

Latest revision as of 01:56, 16 November 2015

Definition

Equivalent definitions in tabular format

No.	Shorthand	A topological space is said to be compact if ...	A topological space $X$ is said to be compact if ...
1	Open cover-finite subcover formulation	every open cover has a finite subcover	for any collection $U_{i},i\in I$ of open subsets of $X$ such that the union of the $U_{i}$ s is $X$ , there is a finite subset $F\subseteq I$ such that $\bigcup _{i\in F}U_{i}=X$ .
2	Finite intersection property formulation	every family of closed sets with the finite intersection property has a nonempty overall intersection	for any collection $C_{i},i\in I$ of closed subsets of $X$ such that every intersection of finitely many of the $C_{i}$ s is nonempty, we also have that the intersection of all $C_{i}$ s is nonempty.
3	Ultrafilter formulation	every ultrafilter of subsets converges to at least one point.	if ${\mathcal {F}}$ is an ultrafilter of subsets of $X$ , there exists $x\in X$ such that ${\mathcal {F}}\to x$ .
4	Subbasis open cover-finite subcover formulation	(fix a choice of subbasis of open subsets) every open cover that uses only members of the subbasis has a finite subcover.	(fix a choice of subbasis of open subsets) for any collection $U_{i},i\in I$ of open subsets of $X$ , all from within the subbasis, such that the union of the $U_{i}$ s is $X$ , there is a finite subset $F\subseteq I$ such that $\bigcup _{i\in F}U_{i}=X$ .

Equivalence of definitions

The equivalence with definition (4) follows from the Alexander subbase theorem.

Examples

In the real line and Euclidean space

Any interval of the form $[a,b]$ (with both $a$ and $b$ real numbers) is a compact space, with the subspace topology inherited from the usual topology on the real line. More generally, any finite union of such intervals is compact.
Compact subsets could look very different from unions of intervals. For instance, the Cantor set is compact.
A subset of the real line, or more generally, of Euclidean space, is compact with the subspace topology if and only if it is closed and bounded (i.e., it can be enclosed inside some large enough ball). See Heine-Borel theorem
Note that it is not true for arbitrary metric spaces that closed and bounded subsets are compact. In fact, for normed real and complex vector spaces, that occur extensively in functional analysis, closed and bounded iff compact is equivalent to being finite-dimensional. Much of the difficulty and challenge of dealing with infinite-dimensional normed real and complex vector spaces is coming up with conditions analogous to compactness that allow reasoning similar to that done in the finite-dimensional case.

More general examples

For a metric space to be compact with the induced topology is equivalent to a condition on it called being totally bounded. See compact metric space.
The geometric realization of any finite simplicial complex is a compact space. (Geometric realizations of simplicial complexes are called polyhedra). See compact polyhedron.
The geometric realiation of a CW-complex with finitely many cells is a compact space. (Geometric realizations of CW-complexes are termed CW-spaces).
Some (but not all) manifolds are compact manifolds, and much of the theory of manifolds relies on the crucial distinction between compact and non-compact manifolds.

In commutative algebra

The spectrum of a commutative unital ring, equipped with the Zariski topology, is always compact (though almost never Hausdorff).

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

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

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

Variations of compactness

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
compact Hausdorff space	compact and Hausdorff: distinct points are separated by disjoint open subsets			Compact T1 space\|FULL LIST, MORE INFO
compact metrizable space	compact and metrizable: arises from a metric space			Compact Hausdorff space\|FULL LIST, MORE INFO
compact manifold	compact and a manifold
compact polyhedron	compact and a polyhedron: arises from a simplicial complex
supercompact space	there is a subbasis of open subsets such that every open cover using the subbasis has a subcover using at most two subsets			\|FULL LIST, MORE INFO
Noetherian space	descending chain of closed subsets stabilizes in finitely many steps	Noetherian implies compact	compact not implies Noetherian	\|FULL LIST, MORE INFO
hereditarily compact space	every subspace is compact		compactness is not hereditary	\|FULL LIST, MORE INFO
finite space	finitely many points			\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
locally compact space	every point is contained in an open subset that's contained in a closed compact subset	compact implies locally compact	locally compact not implies compact	\|FULL LIST, MORE INFO
paracompact space	every open cover has a locally finite open refinement	compact implies paracompact	paracompact not implies compact	\|FULL LIST, MORE INFO
limit point-compact space	every infinite set has a limit point	compact implies limit point-compact	limit point-compact not implies compact	\|FULL LIST, MORE INFO
countably compact space	every countable open cover has a finite subcover	compact implies countably compact	countably compact not implies compact	\|FULL LIST, MORE INFO
sequentially compact space	every infinite sequence has a convergent subsequence	compact implies sequentially compact	sequentially compact not implies compact	\|FULL LIST, MORE INFO
Lindelof space	every open cover has a countable subcover	compact implies Lindelof	Lindelof not implies compact	\|FULL LIST, MORE INFO

Conjunction with other properties

Compact Hausdorff space: Conjunction with the property of being a Hausdorff space
Compact manifold: Conjunction with the property of being a manifold
Compact metrizable space: Conjunction with the property of being a metrizable space

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
product-closed property of topological spaces	Yes	Tychonoff theorem (for arbitrary products). This requires the axiom of choice. For products of finitely many spaces, the proof is simpler and can be deduced from the tube lemma.	If $X_{i},i\in I$ , are all compact spaces, then the product $\prod _{i\in I}X_{i}$ , endowed with the product topology, is also a compact space. Version for two spaces: If $X$ and $Y$ are compact, the product space $X\times Y$ with the product topology is also compact.
weakly hereditary property of topological spaces	Yes	Compactness is weakly hereditary	If $X$ is a compact space and $A$ is a closed subset of $X$ , then $A$ is a compact space with the subspace topology.
subspace-hereditary property of topological spaces	No	Compactness is not subspace-hereditary	It is possible to have a compact space $X$ and a subset $A$ of $X$ such that $A$ is not a compact space with the subspace topology.
fiber-bundle-closed property of topological spaces	Yes	Compactness is fiber bundle-closed	If $p:E\to B$ is a fiber bundle with fiber space $F$ , and both $B$ and $F$ are compact, then $E$ is compact.
continuous image-closed property of topological spaces	Yes	Compactness is continuous image-closed	If $f:X\to Y$ is a continuous map of topological spaces and $X$ is a compact space, then $f(X)$ is a compact space under the subspace topology from $Y$ . In particular, if $f$ is surjective, then $Y$ is compact.
coarsening-preserved property of topological spaces	Yes	Compactness is coarsening-preserved	If a set $X$ is compact under a particular topology, it is also compact under any coarser topology.

Formalisms

Refinement formal expression

In the refinement formalism, the property of compactness has the following refinement formal expression:

Open $\to$ Finite open

References

Textbook references

Topology (2nd edition) by James R. Munkres^{More info}, Page 164 (formal definition)
Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 12 (formal definition)

@@ Line 1: / Line 1: @@
-{{pivotal topospace property}}
+==Definition==
+===Equivalent definitions in tabular format===
-{{basicdef}}
+{| class="sortable" border="1"
+! No. !! Shorthand !! A [[topological space]] is said to be compact if ... !! A [[topological space]] <math>X</math> is said to be compact if ...
+|-
+| 1 || Open cover-finite subcover formulation || every [[open cover]] has a finite subcover || for any collection <math>U_i, i \in I</math> of [[open subset]]s of <math>X</math> such that the union of the <math>U_i</math>s is <math>X</math>, there is a finite subset <math>F \subseteq I</math> such that <math>\bigcup_{i \in F} U_i = X</math>.
+|-
+|2 || Finite intersection property formulation ||every family of closed sets with the [[finite intersection property]] has a nonempty overall intersection || for any collection <math>C_i, i \in I</math> of [[closed subset]]s of <math>X</math> such that every intersection of finitely many of the <math>C_i</math>s is nonempty, we also have that the intersection of all <math>C_i</math>s is nonempty.
+|-
+| 3 || Ultrafilter formulation || every ultrafilter of subsets converges to at least one point. || if <math>\mathcal{F}</math> is an [[ultrafilter of subsets]] of <math>X</math>, there exists <math>x \in X</math> such that <math>\mathcal{F} \to x</math>.
+|-
+| 4 || Subbasis open cover-finite subcover formulation || (fix a choice of [[subbasis]] of [[open subset]]s) every [[open cover]] that uses only members of the subbasis has a finite subcover. || (fix a choice of [[subbasis]] of [[open subset]]s) for any collection <math>U_i, i \in I</math> of [[open subset]]s of <math>X</math>, all from within the subbasis, such that the union of the <math>U_i</math>s is <math>X</math>, there is a finite subset <math>F \subseteq I</math> such that <math>\bigcup_{i \in F} U_i = X</math>.
+|}
-{{surveyarticles|[[:Category:Survey articles related to compactness]]}}
+===Equivalence of definitions===
-==Definition==
+The equivalence with definition (4) follows from the [[Alexander subbase theorem]].
-===Symbol-free definition===
+==Examples==
-A [[topological space]] is said to be '''compact''' if it satisfies the following equivalent conditions:
-* '''Open cover formulation''': Every [[open cover]] has a finite subcover
+===In the real line and Euclidean space===
-* '''Finite intersection property formulation''': Every family of closed sets with the [[finite intersection property]] has a nonempty overall intersection
-* '''Ultrafilter formulation''': Every ultrafilter of subsets converges to at least one point
-===Definition with symbols===
+* Any interval of the form <math>[a,b]</math> (with both <math>a</math> and <math>b</math> real numbers) is a compact space, with the subspace topology inherited from the usual topology on the real line. More generally, any finite union of such intervals is compact.
+* Compact subsets could look very different from unions of intervals. For instance, the [[Cantor set]] is compact.
+* A subset of the real line, or more generally, of Euclidean space, is compact with the subspace topology if and only if it is [[closed subset|closed]] and bounded (i.e., it can be enclosed inside some large enough ball). See [[Heine-Borel theorem]]
+* Note that it is ''not true for arbitrary [[metric space]]s'' that closed and bounded subsets are compact. In fact, for normed real and complex vector spaces, that occur extensively in functional analysis, ''closed and bounded iff compact'' is equivalent to being finite-dimensional. Much of the difficulty and challenge of dealing with infinite-dimensional normed real and complex vector spaces is coming up with conditions analogous to compactness that allow reasoning similar to that done in the finite-dimensional case.
-A [[topological space]] <math>X</math> is said to be '''compact''' if it satisfies the following equivalent condition:
+===More general examples===
-* '''Open cover formulation''': Suppose <math>I</math> is an indexing set and <math>U_i, i \in I</math> is a collection of [[open subset]]s of <math>X</math>, whose union is <math>X</math> (this is the open cover). Then, there exists a finite set <math>F \subset I</math>, such that the union of <math>U_i\ i \in F</math>, is <math>X</math> (this is the finite subcover).
+* For a metric space to be compact with the induced topology is equivalent to a condition on it called being ''totally bounded''. See [[compact metric space]].
-* '''Finite intersection property formulation''': Suppose <math>I</math> is an indexing set and <math>F_i, i \in I</math> is a collection of [[closed subset]]s such that every finite subset has nonempty intersection. Then, the intersection of all <math>F_i</math>s is nonempty.
+* The geometric realization of any finite simplicial complex is a compact space. (Geometric realizations of simplicial complexes are called [[polyhedron|polyhedra]]). See [[compact polyhedron]].
-* '''Ultrafilter formulation''': If <math>\mathcal{F}</math> is an [[ultrafilter of subsets]] of <math>X</math>, there exists <math>x \in X</math> such that <math>\mathcal{F} \to x</math>
+* The geometric realiation of a [[CW-complex]] with finitely many cells is a compact space. (Geometric realizations of CW-complexes are termed [[CW-space]]s).
+* Some (but not all) manifolds are [[compact manifold]]s, and much of the theory of manifolds relies on the crucial distinction between compact and non-compact manifolds.
-==Formalisms==
+===In commutative algebra===
-===Refinement formal expression===
+The [[spectrum of a commutative unital ring]], equipped with the Zariski topology, is always compact (though almost never Hausdorff).
-In the [[refinement]] formalism, the property of compactness has the following [[refinement formal expression]]:
+{{pivotal topospace property}}
-Open <math>\to</math> Finite open
+{{basicdef}}
 ==Relation with other properties==
@@ Line 34: / Line 47: @@
 {{pivotalproperty}}
-* [[:Category:Variations of compactness]]
+* {{#ask: [[variation of::compact space]]|limit = 0|searchlabel = Variations of compactness}}
 ===Stronger properties===
-{| class="wikitable" border="1"
+{| class="sortable" border="1"
-! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::compact Hausdorff space]] || compact and [[Hausdorff space|Hausdorff]]: distinct points are separated by disjoint open subsets || || || {{intermediate notions short|compact space|compact Hausdorff space}}
 |-
-| [[Weaker than::Compact Hausdorff space]] || compact and [[Hausdorff space|Hausdorff]]: distinct points are separated by disjoint open subsets || || || {{intermediate notions short|compact space|compact Hausdorff space}}
+| [[Weaker than::compact metrizable space]] || compact and [[metrizable space|metrizable]]: arises from a [[metric space]] || || || {{intermediate notions short|compact space|compact metrizable space}}
 |-
-| [[Weaker than::Compact metrizable space]] || compact and [[metrizable space|metrizable]]: arises from a [[metric space]] || || || {{intermediate notions short|compact space|compact metrizable space}}
+| [[Weaker than::compact manifold]] || compact and a [[manifold]] || || ||
 |-
-| [[Weaker than::Compact manifold]] || compact and a [[manifold]] || || ||
+| [[Weaker than::compact polyhedron]] || compact and a [[polyhedron]]: arises from a [[simplicial complex]] || || ||
 |-
-| [[Weaker than::Compact polyhedron]] || compact and a [[polyhedron]]: arises from a [[simplicial complex]] || || ||
+| [[Weaker than::supercompact space]] || there is a [[subbasis]] of [[open subset]]s such that every open cover using the subbasis has a subcover using at most two subsets || || || {{intermediate notions short|compact space|supercompact space}}
 |-
 | [[Weaker than::Noetherian space]] || descending chain of closed subsets stabilizes in finitely many steps || [[Noetherian implies compact]] || [[compact not implies Noetherian]] || {{intermediate notions short|compact space|Noetherian space}}
 |-
-| [[Weaker than::Hereditarily compact space]] || every subspace is compact || || [[compactness is not hereditary]] || {{intermediate notions short|compact space|hereditarily compact space}}
+| [[Weaker than::hereditarily compact space]] || every subspace is compact || || [[compactness is not hereditary]] || {{intermediate notions short|compact space|hereditarily compact space}}
 |-
-| [[Weaker than::Finite space]] || finitely many points || || || {{intermediate notions short|compact space|finite space}}
+| [[Weaker than::finite space]] || finitely many points || || || {{intermediate notions short|compact space|finite space}}
 |}
 ===Weaker properties===
-{| class="wikitable" border="1"
+{| class="sortable" border="1"
-! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-| [[Stronger than::Locally compact space]] || every point is contained in an open subset that's contained in a closed compact subset || [[compact implies locally compact]] || [[locally compact not implies compact]] || {{intermediate notions short|locally compact space|compact space}}
+| [[Stronger than::locally compact space]] || every point is contained in an open subset that's contained in a closed compact subset || [[compact implies locally compact]] || [[locally compact not implies compact]] || {{intermediate notions short|locally compact space|compact space}}
 |-
-| [[Stronger than::Paracompact space]] || every [[open cover]] has a locally finite open refinement || [[compact implies paracompact]] || [[paracompact not implies compact]] || {{intermediate notions short|paracompact space|compact space}}
+| [[Stronger than::paracompact space]] || every [[open cover]] has a locally finite open refinement || [[compact implies paracompact]] || [[paracompact not implies compact]] || {{intermediate notions short|paracompact space|compact space}}
 |-
-| [[Stronger than::Limit point-compact space]] || every infinite set has a limit point || [[compact implies limit point-compact]] || [[limit point-compact not implies compact]] || {{intermediate notions short|limit point-compact space|compact space}}
+| [[Stronger than::limit point-compact space]] || every infinite set has a limit point || [[compact implies limit point-compact]] || [[limit point-compact not implies compact]] || {{intermediate notions short|limit point-compact space|compact space}}
 |-
-| [[Stronger than::Countably compact space]] || every countable open cover has a finite subcover || [[compact implies countably compact]] || [[countably compact not implies compact]] || {{intermediate notions short|countably compact space|compact space}}
+| [[Stronger than::countably compact space]] || every countable open cover has a finite subcover || [[compact implies countably compact]] || [[countably compact not implies compact]] || {{intermediate notions short|countably compact space|compact space}}
 |-
-| [[Stronger than::Sequentially compact space]] || every infinite sequence has a convergent subsequence || [[compact implies sequentially compact]] || [[sequentially compact not implies compact]] || {{intermediate notions short|sequentially compact space|compact space}}
+| [[Stronger than::sequentially compact space]] || every infinite sequence has a convergent subsequence || [[compact implies sequentially compact]] || [[sequentially compact not implies compact]] || {{intermediate notions short|sequentially compact space|compact space}}
 |-
 | [[Stronger than::Lindelof space]] || every open cover has a countable subcover || [[compact implies Lindelof]] || [[Lindelof not implies compact]] || {{intermediate notions short|Lindelof space|compact space}}
@@ Line 81: / Line 97: @@
 ==Metaproperties==
-{{DP-closed}}
+{| class="sortable" border="1"
+!Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[satisfies metaproperty::product-closed property of topological spaces]] || Yes || [[Tychonoff theorem]] (for arbitrary products). This requires the axiom of choice. For products of finitely many spaces, the proof is simpler and can be deduced from the [[tube lemma]]. || If <math>X_i, i \in I</math>, are all compact spaces, then the product <math>\prod_{i \in I} X_i</math>, endowed with the [[product topology]], is also a compact space.<br>Version for two spaces: If <math>X</math> and <math>Y</math> are compact, the product space <math>X \times Y</math> with the product topology is also compact.
+|-
+| [[satisfies metaproperty::weakly hereditary property of topological spaces]] || Yes || [[Compactness is weakly hereditary]] || If <math>X</math> is a compact space and <math>A</math> is a [[closed subset]] of <math>X</math>, then <math>A</math> is a compact space with the [[subspace topology]].
+|-
+| [[dissatisfies metaproperty::subspace-hereditary property of topological spaces]] || No || [[Compactness is not subspace-hereditary]] || It is possible to have a compact space <math>X</math> and a subset <math>A</math> of <math>X</math> such that <math>A</math> is ''not'' a compact space with the [[subspace topology]].
+|-
+| [[satisfies metaproperty::fiber-bundle-closed property of topological spaces]] || Yes || [[Compactness is fiber bundle-closed]] || If <math>p:E \to B</math> is a fiber bundle with fiber space <math>F</math>, and both <math>B</math> and <math>F</math> are compact, then <math>E</matH> is compact.
+|-
+| [[satisfies metaproperty::continuous image-closed property of topological spaces]] || Yes || [[Compactness is continuous image-closed]] || If <math>f:X \to Y</math> is a [[continuous map of topological spaces]] and <math>X</matH> is a compact space, then <math>f(X)</math> is a compact space under the [[subspace topology]] from <math>Y</math>. In particular, if <math>f</math> is surjective, then <math>Y</math> is compact.
+|-
+| [[satisfies metaproperty::coarsening-preserved property of topological spaces]] || Yes || [[Compactness is coarsening-preserved]] || If a set <math>X</math> is compact under a particular topology, it is also compact under any [[coarser topology]].
+|}
-Any product of compact spaces is compact. This result is true only in the product topology, not in the box topology. The result is known as the [[Tychonoff theorem]]. For the case of finite direct products, there is a much simpler proof that makes use of the [[tube lemma]].
+==Formalisms==
-{{closed subspace-closed}}
+===Refinement formal expression===
-Any closed subset of a compact space is compact. {{proofat|[[Compactness is weakly hereditary]]}}
+In the [[refinement]] formalism, the property of compactness has the following [[refinement formal expression]]:
-In fact, given any Hausdorff space, every compact subset is closed, so we cannot in general hope for too many compact sets other than the closed ones. (See also [[H-closed space]]).
+Open <math>\to</math> Finite open
-{{coarsening-preserved}}
-Removing open sets reduces the number of possibilities for an open cover, and thus does not damage compactness. In other words, shifting to a coarser topology preserves compactness.
-{{fiber-bundle-closed}}
-The property of being compact is closed under taking fiber bundles; if <math>E</math> is a fiber bundle over base space <math>B</math> with fiber <math>F</math>, and both <math>B</math> and <math>F</math> are compact, so is <math>E</math>.
-{{continuous image-closed}}
-The image of a compact space under a continuous map is again compact. {{proofat|[[Compactness is continuous image-closed]]}}
 ==References==