Stone–Čech compactification (original) (raw)
En mathématiques, et plus précisément en topologie générale, la compactification de Stone-Čech (découverte en 1937 par Marshall Stone et Eduard Čech) est une technique de construction d'un espace compact prolongeant un espace topologique donné X ; plus précisément, il s'agit de la construction d'une application universelle allant de X vers un espace compact βX.
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| dbo:abstract | Die Stone–Čech-Kompaktifizierung ist eine Konstruktion der Topologie zur Einbettung eines topologischen Raumes in einen kompakten Hausdorff-Raum. Die Stone–Čech-Kompaktifizierung eines topologischen Raumes ist der „größte“ kompakte Hausdorff-Raum, der als dichte Teilmenge „enthält“. Präzise ausgedrückt bedeutet das, dass jede Abbildung von in einen kompakten Hausdorff-Raum bezüglich eindeutig faktorisierbar ist. Wenn ein Tychonoff-Raum ist, dann ist die Abbildung eine Einbettung. Man kann sich also als dichten Unterraum von vorstellen. Man benötigt das Auswahlaxiom (etwa in Form des Satzes von Tychonoff), um zu zeigen, dass jeder topologische Raum eine Stone–Čech-Kompaktifizierung besitzt. Auch für sehr einfache Räume ist es sehr schwer, eine konkrete Angabe von zu bekommen. Zum Beispiel ist es unmöglich, einen expliziten Punkt aus anzugeben. Die Stone–Čech-Kompaktifizierung wurde von Marshall Stone (1937) und Eduard Čech (1937) unabhängig voneinander gefunden. Čech stützte sich auf Vorarbeiten von Andrei Nikolajewitsch Tichonow, der 1930 gezeigt hatte, dass jeder vollständig reguläre Raum in ein Produkt von abgeschlossenen Intervallen eingebettet werden kann. Die heute so genannte Stone–Čech-Kompaktifizierung ist dann der Abschluss der Einbettung. Stone betrachtete hingegen den Ring der stetigen, reellwertigen Funktionen auf einem topologischen Raum . Bei seiner Konstruktion ist die heutige Stone–Čech-Kompaktifizierung die Menge der Ultrafilter eines Verbands mit einer bestimmten Topologie. (de) In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX. The Stone–Čech compactification βX of a topological space X is the largest, most general compact Hausdorff space "generated" by X, in the sense that any continuous map from X to a compact Hausdorff space factors through βX (in a unique way). If X is a Tychonoff space then the map from X to its image in βX is a homeomorphism, so X can be thought of as a (dense) subspace of βX; every other compact Hausdorff space that densely contains X is a quotient of βX. For general topological spaces X, the map from X to βX need not be injective. A form of the axiom of choice is required to prove that every topological space has a Stone–Čech compactification. Even for quite simple spaces X, an accessible concrete description of βX often remains elusive. In particular, proofs that βX \ X is nonempty do not give an explicit description of any particular point in βX \ X. The Stone–Čech compactification occurs implicitly in a paper by Andrey Nikolayevich Tychonoff and was given explicitly by Marshall Stone and Eduard Čech. (en) En mathématiques, et plus précisément en topologie générale, la compactification de Stone-Čech (découverte en 1937 par Marshall Stone et Eduard Čech) est une technique de construction d'un espace compact prolongeant un espace topologique donné X ; plus précisément, il s'agit de la construction d'une application universelle allant de X vers un espace compact βX. (fr) 일반위상수학에서 스톤-체흐 콤팩트화(Stone-Čech compact化, 영어: Stone–Čech compactification)는 어떤 위상 공간에 대하여 대응되는 표준적인 콤팩트 하우스도르프 공간이다. 공역이 콤팩트 하우스도르프 공간인 모든 연속 함수는 그 정의역의 스톤-체흐 콤팩트화로 표준적으로 확장시킬 수 있다. (ko) In de topologie, een deelgebied van de wiskunde, is de Stone-Čech-compactificatie een techniek voor de constructie van een universele afbeelding van een topologische ruimte, Xm op een compacte Hausdorff-ruimte βX. De Stone-Čech-compactificatie βX van een topologische ruimte X is de grootste compacte Hausdorff-ruimte die wordt "gegenereerd" door X, in de zin dat elke afbeelding van X op een unieke manier door βX op een compacte Hausdorff-ruimte factoriseert. Als X een Tychonov-ruimte is, dan is de afbeelding van X op haar beeld in βX een homeomorfisme. Hierdoor kan X worden gezien als een (dichte) deelruimte van βX. Voor algemene topologische ruimten X hoeft de afbeelding van X op βX niet injectief te zijn. (nl) Uzwarcenie Čecha-Stone’a – maksymalne (w pewnym, zdefiniowanym niżej sensie) uzwarcenie przestrzeni całkowicie regularnej spełniającej aksjomat oddzielania . Badania nad tego rodzaju uzwarceniami zostały zainicjowane (z odmiennych punktów widzenia) niezależnie przez czeskiego matematyka Eduarda Čecha i amerykańskiego matematyka Marshalla H. Stone’a w 1937. (pl) La compattificazione di Stone-Čech di uno spazio topologico è uno spazio topologico compatto (indicato con ) tale che ogni funzione continua da verso uno spazio topologico compatto può essere estesa ad una funzione definita su tutto . Generalmente, si assume che sia uno spazio di Tychonoff, perché solo in questo caso estende lo spazio di partenza . Fra le varie compattificazioni di uno spazio topologico, quella di Stone-Čech è la "più grande", contrapposta alla compattificazione di Alexandrov, ottenuta aggiungendo un punto solo. (it) Компактификация Стоуна — Чеха (также стоун-чеховская или чех-стоунова компактификация) — максимальная компактификация вполне регулярного топологического пространства. Компактификация Стоуна — Чеха пространства обычно обозначается как . (ru) Компактифікація - Чеха (також стоун-чехівська або чех-стоунова компактифікація) — максимальна компактифікація цілком регулярного топологічного простору. Компактифікація Стоуна - Чеха простору зазвичай позначається як . (uk) 數學的點集拓撲學中，斯通-切赫緊化（Stone–Čech compactification）是構造出從拓撲空間X到緊緻豪斯多夫空間βX的泛映射的技巧。拓撲空間X的斯通－切赫緊化βX是由X「生成」的最大的緊緻豪斯多夫空間，意即任何從X到緊緻豪斯多夫空間的映射，都可以經由βX分解。若X是吉洪諾夫空間，則從X到其在βX中的像的映射是同胚，因此可以想像X是在βX中的稠密子空間。對一般拓撲空間，從X到βX的映射未必是單射。 證明任何拓撲空間都有斯通－切赫緊化，需用到選擇公理的一個形式。即使X是頗簡單的空間，βX通常也是很難以明白具體地描述。譬如βN \ N非空的各種證明（N是自然數集合），都不會直接描述出βN \ N內的任何一點。 (zh) |
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| dbp:authorlink | Eduard Čech (en) Andrey Nikolayevich Tikhonov (en) Marshall Stone (en) |
| dbp:first | Marshall (en) Eduard (en) I.G. (en) Andrey Nikolayevich (en) |
| dbp:last | Stone (en) Čech (en) Tychonoff (en) Koshevnikova (en) |
| dbp:title | Stone-Čech compactification (en) |
| dbp:txt | yes (en) |
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| dbp:year | 1930 (xsd:integer) 1937 (xsd:integer) |
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| gold:hypernym | dbr:Technique |
| rdf:type | owl:Thing dbo:TopicalConcept |
| rdfs:comment | En mathématiques, et plus précisément en topologie générale, la compactification de Stone-Čech (découverte en 1937 par Marshall Stone et Eduard Čech) est une technique de construction d'un espace compact prolongeant un espace topologique donné X ; plus précisément, il s'agit de la construction d'une application universelle allant de X vers un espace compact βX. (fr) 일반위상수학에서 스톤-체흐 콤팩트화(Stone-Čech compact化, 영어: Stone–Čech compactification)는 어떤 위상 공간에 대하여 대응되는 표준적인 콤팩트 하우스도르프 공간이다. 공역이 콤팩트 하우스도르프 공간인 모든 연속 함수는 그 정의역의 스톤-체흐 콤팩트화로 표준적으로 확장시킬 수 있다. (ko) Uzwarcenie Čecha-Stone’a – maksymalne (w pewnym, zdefiniowanym niżej sensie) uzwarcenie przestrzeni całkowicie regularnej spełniającej aksjomat oddzielania . Badania nad tego rodzaju uzwarceniami zostały zainicjowane (z odmiennych punktów widzenia) niezależnie przez czeskiego matematyka Eduarda Čecha i amerykańskiego matematyka Marshalla H. Stone’a w 1937. (pl) La compattificazione di Stone-Čech di uno spazio topologico è uno spazio topologico compatto (indicato con ) tale che ogni funzione continua da verso uno spazio topologico compatto può essere estesa ad una funzione definita su tutto . Generalmente, si assume che sia uno spazio di Tychonoff, perché solo in questo caso estende lo spazio di partenza . Fra le varie compattificazioni di uno spazio topologico, quella di Stone-Čech è la "più grande", contrapposta alla compattificazione di Alexandrov, ottenuta aggiungendo un punto solo. (it) Компактификация Стоуна — Чеха (также стоун-чеховская или чех-стоунова компактификация) — максимальная компактификация вполне регулярного топологического пространства. Компактификация Стоуна — Чеха пространства обычно обозначается как . (ru) Компактифікація - Чеха (також стоун-чехівська або чех-стоунова компактифікація) — максимальна компактифікація цілком регулярного топологічного простору. Компактифікація Стоуна - Чеха простору зазвичай позначається як . (uk) 數學的點集拓撲學中，斯通-切赫緊化（Stone–Čech compactification）是構造出從拓撲空間X到緊緻豪斯多夫空間βX的泛映射的技巧。拓撲空間X的斯通－切赫緊化βX是由X「生成」的最大的緊緻豪斯多夫空間，意即任何從X到緊緻豪斯多夫空間的映射，都可以經由βX分解。若X是吉洪諾夫空間，則從X到其在βX中的像的映射是同胚，因此可以想像X是在βX中的稠密子空間。對一般拓撲空間，從X到βX的映射未必是單射。 證明任何拓撲空間都有斯通－切赫緊化，需用到選擇公理的一個形式。即使X是頗簡單的空間，βX通常也是很難以明白具體地描述。譬如βN \ N非空的各種證明（N是自然數集合），都不會直接描述出βN \ N內的任何一點。 (zh) Die Stone–Čech-Kompaktifizierung ist eine Konstruktion der Topologie zur Einbettung eines topologischen Raumes in einen kompakten Hausdorff-Raum. Die Stone–Čech-Kompaktifizierung eines topologischen Raumes ist der „größte“ kompakte Hausdorff-Raum, der als dichte Teilmenge „enthält“. Präzise ausgedrückt bedeutet das, dass jede Abbildung von in einen kompakten Hausdorff-Raum bezüglich eindeutig faktorisierbar ist. Wenn ein Tychonoff-Raum ist, dann ist die Abbildung eine Einbettung. Man kann sich also als dichten Unterraum von vorstellen. (de) In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX. The Stone–Čech compactification βX of a topological space X is the largest, most general compact Hausdorff space "generated" by X, in the sense that any continuous map from X to a compact Hausdorff space factors through βX (in a unique way). If X is a Tychonoff space then the map from X to its image in βX is a homeomorphism, so X can be thought of as a (dense) subspace of βX; every other compact Hausdorff space that densely contains X is a quotient of βX. For general topological spaces X, the map from X to βX need not be injective. (en) In de topologie, een deelgebied van de wiskunde, is de Stone-Čech-compactificatie een techniek voor de constructie van een universele afbeelding van een topologische ruimte, Xm op een compacte Hausdorff-ruimte βX. De Stone-Čech-compactificatie βX van een topologische ruimte X is de grootste compacte Hausdorff-ruimte die wordt "gegenereerd" door X, in de zin dat elke afbeelding van X op een unieke manier door βX op een compacte Hausdorff-ruimte factoriseert. (nl) |
| rdfs:label | Stone-Čech-Kompaktifizierung (de) Compactification de Stone-Čech (fr) Compattificazione di Stone-Čech (it) 스톤-체흐 콤팩트화 (ko) Stone-Čech-compactificatie (nl) Uzwarcenie Čecha-Stone’a (pl) Stone–Čech compactification (en) Компактификация Стоуна — Чеха (ru) Компактифікація Стоуна — Чеха (uk) 斯通-切赫緊化 (zh) |
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