Compact space: Difference between revisions

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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

For survey articles related to this, refer: Category:Survey articles related to compactness

Definition

Symbol-free definition

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
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

A topological space $X$ is said to be compact if it satisfies the following equivalent condition:

Open cover formulation: Suppose $I$ is an indexing set and $U_{i}, i \in I$ is a collection of open subsets of $X$ , whose union is $X$ (this is the open cover). Then, there exists a finite set $F \subset I$ , such that the union of $U_{i} i \in F$ , is $X$ (this is the finite subcover).
Finite intersection property formulation: Suppose $I$ is an indexing set and $F_{i}, i \in I$ is a collection of closed subsets such that every finite subset has nonempty intersection. Then, the intersection of all $F_{i}$ s is nonempty.
Ultrafilter formulation: If $F$ is an ultrafilter of subsets of $X$ , there exists $x \in X$ such that $F \to x$

Formalisms

Refinement formal expression

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

Open $\to$ Finite open

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

Category:Variations of compactness

Stronger properties

property	quick description	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			Template:Intermediate notions
Compact metrizable space	compact and metrizable: arises from a metric space			Compact Hausdorff space\|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	quick description	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

Products

This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products

Any product of compact spaces is compact. This result is true only in theproduct 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.

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

Any closed subset of a compact space is compact. For full proof, refer: Compactness is weakly hereditary

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).

Coarsening

This property of topological spaces is preserved under coarsening, viz, if a set with a given topology has the property, the same set with a coarser topology also has the property

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 bundles

This property of topological spaces is a fiber bundle-closed property of topological spaces: it is closed under taking fiber bundles, viz if the base space and fiber both satisfy the given property, so does the total space.
Manifold, Orientable manifold

The property of being compact is closed under taking fiber bundles; if $E$ is a fiber bundle over base space $B$ with fiber $F$ , and both $B$ and $F$ are compact, so is $E$ .

Closure under continuous images

The image, via a continuous map, of a topological space having this property, also has this property

The image of a compact space under a continuous map is again compact.

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 37: / Line 37: @@
 ===Stronger properties===
+{| class="wikitable" border="1"
+! property !! quick description !! 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|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::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::Finite space]] || finitely many points || || || {{intermediate notions short|compact space|finite space}}
+|}
+===Weaker properties===
-* [[Compact Hausdorff space]]
+{| class="wikitable" border="1"
-* [[Compact metrizable space]]
+! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions !! comparison
-* [[Noetherian space]]
+|-
-* [[Hereditarily 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}}
-* [[Irreducible 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}}
-===Weaker properties===
+|-
+| [[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::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}}
+|}
-* [[Locally compact space]]
-* [[Paracompact space]]
-* [[Limit point-compact space]]
-* [[Countably compact space]]
 ===Conjunction with other properties===