General topology Compactness Concept
Compact space
Summary
A compact space is a topological space in which each open cover has a finite subcover. Closed bounded intervals of the real line are the basic examples.
Where this leads
102 pages build on this one.
- Krylov–Bogolyubov theorem
- Long line
- Stone–Weierstrass theorem
- Ergodic decomposition
- Holomorphic functions on compact complex manifolds are constant
- Perfect map
- Compactification
- L2 convergence of Fourier series
- Brouwer's characterization of the Cantor space
- Cartan–Hadamard theorem
- Neural network approximation
- Proper holomorphic map
- Limit point compact space
- Particular point topology
- Compact group
- Proper group action
- Cartan–Iwasawa–Malcev theorem
- Chow's theorem
- Preissmann's theorem
- Stone space
- Table of Lie groups
- Hawaiian earring
- Space of functions vanishing at infinity
- Fort space
- Complex projective space
- Generalized Poincaré conjecture
- Excluded point topology
- Property (T)
- Continuous functional calculus
- Shift space
- Monad in nonstandard topology
- Willmore energy
- Hochster's theorem on spectral spaces
- Haar's uniqueness theorem
- Gleason's theorem on projective compact spaces
- Conley's fundamental theorem of dynamical systems
- Tychonoff plank
- Uniform algebra
- Hopf conjecture
- Compactness via ultrafilters
- Countably compact space
- Priestley space
- Haagerup property
- Attractor
- Compact Lie group
- Maximal compact subgroup
- Dynamical billiards
- Approximation of a kernel by degenerate kernels
- Lefschetz fixed-point theorem
- Algebraic dimension
- Crystallographic group
- Calabi–Eckmann manifold
- Compactly generated space
- Continuum
- Variational principle for topological entropy
- Kolmogorov criterion for best approximation
- Minimal set
- Chebyshev system
- Quasi-compactness of the spectrum
- Fundamental class
- Density of trigonometric polynomials in continuous functions
- Unique ergodicity
- Alexander duality
- Topological entropy
- Compactness of integral operators
- Stone–Čech compactification
- Cartan–Serre finiteness theorem
- Grauert's semicontinuity theorem
- Brouwer fixed-point theorem
- Alexander subbase theorem
- Commutative Gelfand–Naimark theorem
- Fourier basis of L2 of the circle
- Equicontinuous system
- Jentzsch's theorem on positive kernels
- Sierpiński carpet
- Kneser's theorem on solution sets
- Hahn–Mazurkiewicz theorem
- Banach–Stone theorem
- Hopf–Rinow theorem
- Spectral space
- Noncommutative geometry
- Equivariant index theorem
- Systolic geometry
- Gromov–Hausdorff convergence
- Étale fundamental group
- Basel problem via Fourier series
- Profinite group
- Cantor algebra
- Quasi-compact morphism
- Properly discontinuous action
- G-signature theorem
- Tube lemma
- Cantor space
- Rate function
- Menger sponge
- Convergence along an ultrafilter
- Algorithmic randomness
- Choquet capacity
- Compact quantum group
- Myers–Steenrod theorem
- Reeb stability theorem
- Grauert's direct image theorem
Further reading
- Web https://en.wikipedia.org/wiki/Compact_space, lead section
About this page
| Tag | 0TVJZYB6SH |
|---|---|
| URI | m2://0TVJZYB6SH |
| Kind | concept |
| Writing | W1 stated |
| MSC | 54D30 |
| Specializes | Topological space |
| Same as |
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What this page needs for W2
- edge not_instance_of in: at least one non-example (0 of 1)
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What links here
Instances
Definitions
Used by
- Krylov–Bogolyubov theorem
- Long line
- Stone–Weierstrass theorem
- Ergodic decomposition
- Holomorphic functions on compact complex manifolds are constant
- Perfect map
- Compactification
- Brouwer's characterization of the Cantor space
- Neural network approximation
- Proper holomorphic map
- Limit point compact space
- Particular point topology
- Compact group
- Proper group action
- Cartan–Iwasawa–Malcev theorem
- Stone space
- Table of Lie groups
- Hawaiian earring
- Space of functions vanishing at infinity
- Fort space
- Complex projective space
- Generalized Poincaré conjecture
- Property (T)
- Shift space
- Monad in nonstandard topology
- Willmore energy
- Hochster's theorem on spectral spaces
- Haar's uniqueness theorem
- Gleason's theorem on projective compact spaces
- Conley's fundamental theorem of dynamical systems
- Tychonoff plank
- Uniform algebra
- Hopf conjecture
- Compactness via ultrafilters
- Countably compact space
- Priestley space
- Haagerup property
- Attractor
- Compact Lie group
- Maximal compact subgroup
- Dynamical billiards
- Lefschetz fixed-point theorem
- Algebraic dimension
- Crystallographic group
- Compactly generated space
- Continuum
- Variational principle for topological entropy
- Kolmogorov criterion for best approximation
- Minimal set
- Chebyshev system
- Quasi-compactness of the spectrum
- Fundamental class
- Unique ergodicity
- Alexander duality
- Topological entropy
- Stone–Čech compactification
- Cartan–Serre finiteness theorem
- Alexander subbase theorem
- Equicontinuous system
- Sierpiński carpet
- Hahn–Mazurkiewicz theorem
- Banach–Stone theorem
- Hopf–Rinow theorem
- Spectral space
- Systolic geometry
- Gromov–Hausdorff convergence
- Profinite group
- Quasi-compact morphism
- Properly discontinuous action
- Tube lemma
- Cantor space
- Rate function
- Menger sponge
- Convergence along an ultrafilter
- Choquet capacity
- Compact quantum group
- Myers–Steenrod theorem
- Reeb stability theorem