Quotient space (topology) (original) (raw)
En matemàtiques, la topologia quocient és una topologia definida sobre el conjunt quocient generat per una relació d'equivalència sobre un espai topològic.
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| dbo:abstract | En matemàtiques, la topologia quocient és una topologia definida sobre el conjunt quocient generat per una relació d'equivalència sobre un espai topològic. (ca) Die Quotiententopologie (auch Identifizierungstopologie genannt) ist ein Begriff aus dem mathematischen Teilgebiet der Topologie. Anschaulich entsteht diese Topologie, wenn man Punkte „zusammenklebt“, d. h. zwei ehemals verschiedene Punkte als ein und denselben Punkt identifiziert. Solche Punkte werden mittels Äquivalenzrelationen festgelegt. Das geschieht im Allgemeinen, um neue topologische Räume aus bestehenden abzuleiten. Zu einer Verallgemeinerung dieser Konstruktion vergleiche den Artikel Finaltopologie. (de) En matemáticas, la topología cociente consiste intuitivamente en crear una topología pegando ciertos puntos sobre otros, en un espacio dado, por medio de una relación de equivalencia bien definida. El nuevo espacio así generado recibe el nombre de espacio cociente. Ejemplos conocidos son el toro matemático o la banda de Möbius. (es) In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space. Intuitively speaking, the points of each equivalence class are identified or "glued together" for forming a new topological space. For example, identifying the points of a sphere that belong to the same diameter produces the projective plane as a quotient space. (en) En mathématiques, la topologie quotient consiste intuitivement à créer une topologie en collant certains points d'un espace donné sur d'autres, par le biais d'une relation d'équivalence bien choisie. Cela est souvent fait dans le but de construire de nouveaux espaces à partir d'anciens. On parle alors d'espace topologique quotient. (fr) ( 벡터 공간의 몫공간에 대해서는 몫 벡터 공간 문서를 참고하십시오.) 일반위상수학에서 몫공간(-空間, 영어: quotient space)은 어떤 위상 공간의 몫집합 위에 표준적으로 존재하는 위상 공간이다. (ko) 位相空間論およびそれに関連する数学の各分野において、等化空間（とうかくうかん、英: identification space）または商位相空間（しょういそうくうかん、英: quotient topological space）あるいは単に商空間 (quotient space) とは、直観的には与えられた空間のある種の点の集まりを「貼合せ」("gluing together") あるいは同一視してしまうことによって得られる新しい空間である。ただし、ここで貼合わせられるべき点の集まりというのは、何らかの同値関係によって決定される。 このような商空間構成は、与えられた位相空間から新たな空間を構成する方法の一つとして広く用いられる。 (ja) In topologia, la topologia quoziente è intuitivamente quella ottenuta da uno spazio topologico "attaccando" alcuni punti fra loro. Lo spazio topologico che si ottiene viene anche chiamato spazio quoziente. (it) In de topologie, een deelgebied van de wiskunde, is een quotiënttopologie de geïnduceerde topologie op de equivalentieklassen van een equivalentierelatie op een topologische ruimte. Er ontstaat een nieuwe topologiche ruimte van de "aan elkaar geplakte" equivalente elementen. (nl) Topologia ilorazowa – w topologii, dziale matematyki, najbogatsza topologia określona na zbiorze ilorazowym, wyznaczonym przez relację równoważności określoną na danej przestrzeni topologicznej, względem której odwzorowanie ilorazowe jest ciągłe. Szczególne przypadki topologii ilorazowych badali jako pierwsi Robert Lee Moore oraz Paweł Aleksandrow. (pl) Em topologia, um espaço topológico quociente, X, é definido como, dado uma relação de equivalência, ~, o espaço topológico \(([X], au)\), onde \([X]\) denota as classes de equivalencia de X e au={U \subset 2^[X]| \união_{[x] \in U} x é aberto em X}. O quociente de um espaço topológico X por uma relação de equivalência ~ é o conjunto X/~ das classes de equivalência munido da topologia (chamada topologia quociente) cujos abertos são os conjuntos de classes cuja reunião é um aberto de X. (pt) 在拓扑学及其相关数学领域，一个商空间（quotient space，也称为等化空间identification space）直观上说是将一个给定空间的一些点等同或“黏合在一起”；由一个等价关系确定哪些点是等同的。这是从给定空间构造新空间的常见方法。 (zh) Фактор-простір — простір класів еквівалентності топологічного простору за заданим відношенням еквівалентності. (uk) |
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| rdfs:comment | En matemàtiques, la topologia quocient és una topologia definida sobre el conjunt quocient generat per una relació d'equivalència sobre un espai topològic. (ca) Die Quotiententopologie (auch Identifizierungstopologie genannt) ist ein Begriff aus dem mathematischen Teilgebiet der Topologie. Anschaulich entsteht diese Topologie, wenn man Punkte „zusammenklebt“, d. h. zwei ehemals verschiedene Punkte als ein und denselben Punkt identifiziert. Solche Punkte werden mittels Äquivalenzrelationen festgelegt. Das geschieht im Allgemeinen, um neue topologische Räume aus bestehenden abzuleiten. Zu einer Verallgemeinerung dieser Konstruktion vergleiche den Artikel Finaltopologie. (de) En matemáticas, la topología cociente consiste intuitivamente en crear una topología pegando ciertos puntos sobre otros, en un espacio dado, por medio de una relación de equivalencia bien definida. El nuevo espacio así generado recibe el nombre de espacio cociente. Ejemplos conocidos son el toro matemático o la banda de Möbius. (es) En mathématiques, la topologie quotient consiste intuitivement à créer une topologie en collant certains points d'un espace donné sur d'autres, par le biais d'une relation d'équivalence bien choisie. Cela est souvent fait dans le but de construire de nouveaux espaces à partir d'anciens. On parle alors d'espace topologique quotient. (fr) ( 벡터 공간의 몫공간에 대해서는 몫 벡터 공간 문서를 참고하십시오.) 일반위상수학에서 몫공간(-空間, 영어: quotient space)은 어떤 위상 공간의 몫집합 위에 표준적으로 존재하는 위상 공간이다. (ko) 位相空間論およびそれに関連する数学の各分野において、等化空間（とうかくうかん、英: identification space）または商位相空間（しょういそうくうかん、英: quotient topological space）あるいは単に商空間 (quotient space) とは、直観的には与えられた空間のある種の点の集まりを「貼合せ」("gluing together") あるいは同一視してしまうことによって得られる新しい空間である。ただし、ここで貼合わせられるべき点の集まりというのは、何らかの同値関係によって決定される。 このような商空間構成は、与えられた位相空間から新たな空間を構成する方法の一つとして広く用いられる。 (ja) In topologia, la topologia quoziente è intuitivamente quella ottenuta da uno spazio topologico "attaccando" alcuni punti fra loro. Lo spazio topologico che si ottiene viene anche chiamato spazio quoziente. (it) In de topologie, een deelgebied van de wiskunde, is een quotiënttopologie de geïnduceerde topologie op de equivalentieklassen van een equivalentierelatie op een topologische ruimte. Er ontstaat een nieuwe topologiche ruimte van de "aan elkaar geplakte" equivalente elementen. (nl) Topologia ilorazowa – w topologii, dziale matematyki, najbogatsza topologia określona na zbiorze ilorazowym, wyznaczonym przez relację równoważności określoną na danej przestrzeni topologicznej, względem której odwzorowanie ilorazowe jest ciągłe. Szczególne przypadki topologii ilorazowych badali jako pierwsi Robert Lee Moore oraz Paweł Aleksandrow. (pl) Em topologia, um espaço topológico quociente, X, é definido como, dado uma relação de equivalência, ~, o espaço topológico \(([X], au)\), onde \([X]\) denota as classes de equivalencia de X e au={U \subset 2^[X]| \união_{[x] \in U} x é aberto em X}. O quociente de um espaço topológico X por uma relação de equivalência ~ é o conjunto X/~ das classes de equivalência munido da topologia (chamada topologia quociente) cujos abertos são os conjuntos de classes cuja reunião é um aberto de X. (pt) 在拓扑学及其相关数学领域，一个商空间（quotient space，也称为等化空间identification space）直观上说是将一个给定空间的一些点等同或“黏合在一起”；由一个等价关系确定哪些点是等同的。这是从给定空间构造新空间的常见方法。 (zh) Фактор-простір — простір класів еквівалентності топологічного простору за заданим відношенням еквівалентності. (uk) In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space. (en) |
| rdfs:label | Topologia quocient (ca) Quotiententopologie (de) Topología cociente (es) Topologie quotient (fr) Topologia quoziente (it) 商位相空間 (ja) 몫공간 (ko) Quotiënttopologie (nl) Topologia ilorazowa (pl) Quotient space (topology) (en) Espaço topológico quociente (pt) Факторпространство (ru) 商空间 (zh) Фактор-простір (uk) |
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