Compact space
From Topospaces
Definition
Equivalent definitions in tabular format
No. | Shorthand | A topological space is said to be compact if ... | A topological space is said to be compact if ... |
---|---|---|---|
1 | Open cover-finite subcover formulation | every open cover has a finite subcover | for any collection of open subsets of such that the union of the s is , there is a finite subset such that . |
2 | Finite intersection property formulation | every family of closed sets with the finite intersection property has a nonempty overall intersection | for any collection of closed subsets of such that every intersection of finitely many of the s is nonempty, we also have that the intersection of all s is nonempty. |
3 | Ultrafilter formulation | every ultrafilter of subsets converges to at least one point. | if is an ultrafilter of subsets of , there exists such that . |
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 of open subsets of , all from within the subbasis, such that the union of the s is , there is a finite subset such that . |
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 (with both and 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
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 | | | ||
Noetherian space | descending chain of closed subsets stabilizes in finitely many steps | Noetherian implies compact | compact not implies Noetherian | | |
hereditarily compact space | every subspace is compact | compactness is not hereditary | | | |
finite space | finitely many points | | |
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 | | |
paracompact space | every open cover has a locally finite open refinement | compact implies paracompact | paracompact not implies compact | | |
limit point-compact space | every infinite set has a limit point | compact implies limit point-compact | limit point-compact not implies compact | | |
countably compact space | every countable open cover has a finite subcover | compact implies countably compact | countably compact not implies compact | | |
sequentially compact space | every infinite sequence has a convergent subsequence | compact implies sequentially compact | sequentially compact not implies compact | | |
Lindelof space | every open cover has a countable subcover | compact implies Lindelof | Lindelof not implies compact | | |
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 , are all compact spaces, then the product , endowed with the product topology, is also a compact space. Version for two spaces: If and are compact, the product space with the product topology is also compact. |
weakly hereditary property of topological spaces | Yes | Compactness is weakly hereditary | If is a compact space and is a closed subset of , then 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 and a subset of such that is not a compact space with the subspace topology. |
fiber-bundle-closed property of topological spaces | Yes | Compactness is fiber bundle-closed | If is a fiber bundle with fiber space , and both and are compact, then is compact. |
continuous image-closed property of topological spaces | Yes | Compactness is continuous image-closed | If is a continuous map of topological spaces and is a compact space, then is a compact space under the subspace topology from . In particular, if is surjective, then is compact. |
coarsening-preserved property of topological spaces | Yes | Compactness is coarsening-preserved | If a set 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 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)