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Stone–Čech compactification

An important problem in topology is how to enlarge a space by adding points so that certain kinds of limits exist. The Stone–Čech compactification of a space provides the most extensive such enlargement: it adds enough points to ensure the existence of all generalized limits, including those detected by nets or ultrafilters rather than ordinary…

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The Stone–Čech compactification of the natural numbers

RemainderDiscrete topologySet-theoretic topologyParovicenko's theoremsContinuum hypothesis
28 related topics

Constructions

Kan extensionProper classPower setUnit intervalUltrafilters
19 related topics

Universal property and functoriality

Continuous mapUp toHomeomorphism§ ConstructionsLocally compact
12 related topics

Applications

Ramsey theoryIdempotentIdealZorn's lemmaVan der Waerden's theorem
8 related topics

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Universal property and functoriality

Examples

Constructions

The Stone–Čech compactification of the natural numbers

Applications

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Stone–Čech compactification

Nodes104
Edges103
Triples145
Avg. degree1.98
Density0.019231
Components1

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Stone–Čech compactification

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related to References · 74
Stone–Čech compactification → Algebra, Aliprantis, Allen, American Mathematical Society, Andrey, Annals, Applications, Beckenstein, Berlin, BF01782364, BF03025901, Boca Raton, Boolean, Border, Cech, Charalambos, Co, Compactification, CRC Press, De Gruyter Expositions
has application · 12
Stone–Čech compactification → Central Sets Theorem, Furstenberg, Hindman, In, Ramsey, Stone, Strauss, The, The Central Sets Theorem, The Stone, This, While
related to Construction using products · 9
Stone–Čech compactification → By Tychonoff's, CHaus, Hausdorff, Kan, One, Stone, There, This, Top
related to Extension of continuous functions · 8
Stone–Čech compactification → Because, For, However, If, In, Stone, The Stone, This
related to An application: the dual space of the space of bounded sequences of reals · 7
Stone–Čech compactification → According, Banach, Given, Hausdorff, Since, The Stone, This
related to history · 7
Stone–Čech compactification → Andrey Nikolayevich Tikhonov, Despite Tychonoff's, Hausdorff, In, Stone, Tychonoff, Tychonoff's
related to Comparison with one-point compactification · 6
Stone–Čech compactification → Another, By, For, In, It, Stone
related to Examples · 6
Stone–Čech compactification → Hausdorff, If, Stone, The Stone, These, Tychonoff
related to Construction using C*-algebras · 5
Stone–Čech compactification → C0, Cb, Here Cb, Notice, The Stone
related to External links · 5
Stone–Čech compactification → Compactness, Planet MathDror Bar-Natan, Stone, Stone-Čech Compactification, Ultrafilters

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Important terminology

compactification space stone displaystyle čech hausdorff compact beta continuous ultrafilters spaces βx map set βn construction topology topological universal functions

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SubjectPredicateObjectConfidenceSrc
Stone–Čech compactificationis atechnique for constructing a universal map from a topological space X to a compact Hausdorff space β X0.90text
Van der Waerden's theoreminstance ofThis topological approach provides proofs for results0.80text
the Halesinstance ofThis topological approach provides proofs for results0.80text
Stone–Čech compactificationhas applicationThe Stone0.60section
Stone–Čech compactificationhas applicationRamsey0.60section
Stone–Čech compactificationhas applicationWhile0.60section
Stone–Čech compactificationhas applicationThis0.60section
Stone–Čech compactificationhas applicationThe0.60section
Stone–Čech compactificationhas applicationCentral Sets Theorem0.60section
Stone–Čech compactificationhas applicationIn0.60section
Stone–Čech compactificationhas applicationStone0.60section
Stone–Čech compactificationhas applicationThe Central Sets Theorem0.60section

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