Compact space

Definition

No.	Shorthand	A topological space is said to be compact if ...	A topological space $X$ is said to be compact if ...
1	Open cover 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 $\mathcal{F}\to x$.

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.
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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).
• 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.
• The geometric realization of any finite simplicial complex is a compact space. (Geometric realizations of simplicial complexes are called polyhedra).
• The geometric realiation of a CW-complex with finitely many cells is a compact space. (Geometric realizations of CW-complexes are termed CW-spaces).

In commutative algebra

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

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

• 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
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	Section in this article
product-closed property of topological spaces	Yes	Tychonoff theorem	#Products
weakly hereditary property of topological spaces	Yes	Compactness is weakly hereditary	#Weak hereditariness
subspace-hereditary property of topological spaces	No	Compactness is not subspace-hereditary	#Hereditariness
fiber-bundle-closed property of topological spaces	Yes	Compactness is fiber bundle-closed	#Fiber bundles
continuous image-closed property of topological spaces	Yes	Compactness is continuous image-closed	#Closure under continuous images
coarsening-preserved property of topological spaces	Yes	Compactness is coarsening-preserved	#Coarsening

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 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.

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. For full proof, refer: Compactness is coarsening-preserved

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. For full proof, refer: Compactness is continuous image-closed

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)