Stone–Čech compactification
E679313
The Stone–Čech compactification is a construction in topology that associates to any topological space a universal, maximally extensive compact Hausdorff space into which it densely embeds.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
compactification
ⓘ
functor ⓘ topological construction ⓘ universal construction ⓘ |
| alsoKnownAs | Stone–Čech remainder construction NERFINISHED ⓘ |
| appliesTo |
Tychonoff spaces
NERFINISHED
ⓘ
completely regular spaces ⓘ |
| area |
category theory
ⓘ
general topology ⓘ |
| characterization |
can be described as the maximal ideal space of C_b(X)
ⓘ
can be described via ultrafilters on X ⓘ |
| codomain | compact Hausdorff spaces ⓘ |
| constructionMethod |
via maximal ideals of C_b(X)
ⓘ
via ultrafilters on the underlying set of X ⓘ |
| constructionOf | a compact Hausdorff space βX containing X densely ⓘ |
| domain | topological spaces ⓘ |
| feature |
βX contains X as a dense subspace
ⓘ
βX is Hausdorff ⓘ βX is compact ⓘ βX is extremally disconnected for certain X ⓘ βX is unique up to homeomorphism ⓘ |
| field | topology ⓘ |
| hasComponent | Stone–Čech remainder βX\X NERFINISHED ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| maps | a continuous map f:X→Y to a continuous map βf:βX→βY ⓘ |
| minimalityProperty | βX is the largest compactification of X in the sense of continuous maps ⓘ |
| namedAfter |
Eduard Čech
NERFINISHED
ⓘ
Marshall Harvey Stone NERFINISHED ⓘ |
| preserves | products up to natural homeomorphism for Tychonoff spaces ⓘ |
| property |
extends continuous maps uniquely
ⓘ
gives a dense embedding of the original space ⓘ is functorial on the category of topological spaces ⓘ is universal among compact Hausdorff extensions ⓘ |
| relatedConcept |
Alexandroff one-point compactification
NERFINISHED
ⓘ
C*-algebra of continuous bounded functions ⓘ Gelfand duality NERFINISHED ⓘ ultrafilter ⓘ Čech–Stone compactification NERFINISHED ⓘ |
| specialCase | one-point compactification for certain locally compact spaces ⓘ |
| symbol | βX ⓘ |
| universalProperty | every continuous map from X to a compact Hausdorff space factors uniquely through βX ⓘ |
| usedIn |
Ramsey theory
NERFINISHED
ⓘ
functional analysis ⓘ nonstandard analysis ⓘ semigroup theory ⓘ set-theoretic topology ⓘ topological dynamics ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.