Compact space

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.
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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)