View source for Compact space

==Definition==

===Equivalent definitions in tabular format===

{| 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>.
|}

===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 <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.

===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 [[polyhedron|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-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.

===In commutative algebra===

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

{{pivotal topospace property}}

{{basicdef}}

==Relation with other properties==

{{pivotalproperty}}

* {{#ask: [[variation of::compact space]]|limit = 0|searchlabel = Variations of compactness}}

===Stronger properties===
{| class="sortable" border="1"
! 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 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 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::finite space]] || finitely many points || || || {{intermediate notions short|compact space|finite space}}
|}

===Weaker properties===

{| class="sortable" border="1"
! 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::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::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}}
|}


===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==

{| 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]].
|}

==Formalisms==

===Refinement formal expression===

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

Open <math>\to</math> Finite open


==References==
===Textbook references===
* {{booklink|Munkres}}, Page 164 (formal definition)
* {{booklink|SingerThorpe}}, Page 12 (formal definition)