identification of points that are equivalent to one another in some equivalence relation. (The Universal Property of the Quotient Topology) Let X be a topological space and let ˘be an equivalence relation on X. Endow the set X=˘with the quotient topology and let ˇ: X!X=˘be the canonical surjection. The space obtained is called a quotient space and is denoted V/N (read V mod N or V by N).. Quotient spaces Theorem 4 (above) will be combined with the bijective correspondence between sub-σ-fields, measure subalgebras and linear sublattices described in the corresponding section of "Measure space".. How to use quotient in a sentence. In Section 2 we recall all necessary definitions, and in Section 3 we consider two axioms, denoted by M and G, each not derivable from S4 and the other one, and for each of them we give necessary and sufficient conditions under which it is valid in a quotient space of a finite CW-complex, a particular point topological space, and an excluded point topological space. You can have quotient spaces in set theory, group theory, field theory, linear algebra, topology, and others. Math. When we have a group G acting on a space X, there is a “natural” quotient space. Illustrated definition of Quotient: The answer after we divide one number by another. Quotient of a Banach space by a subspace. quotient-space definition: Noun 1. attributive form of quotient spacequotient-space mapNoun (plural quotient spaces) 2. Theorem 5.1. \begin{align} \quad \| (x_{n_2} + y_2) - (x_{n_3} + y_3) \| \leq \| (x_{n_2} - x_{n_3}) + M \| + \frac{1}{4} < \frac{1}{4} + \frac{1}{4} = \frac{1}{2} \end{align} “Quotient space” covers a lot of ground. quotient space: Meaning and Definition of. A topological space is sequential if and only if it is a quotient of a metric space. In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space.The points to be identified are specified by an equivalence relation. This is commonly done in order to construct new spaces from given ones. If X is a topological space and A is a set and if : → is a surjective map, then there exist exactly one topology on A relative to which f is a quotient map; it is called the quotient topology induced by f . n. The number obtained by dividing one quantity by another. View each of these “orbit” sets as a single point in some new space X∗. In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. Definition.Let (X, S) be a topological space, let Q be a set, and let π : X → Q be a surjective mapping.The resulting quotient topology (or identification topology) on Q is defined to be See more. [4] Generalizations of metric spaces This is commonly done in order to construct new spaces from given ones. quotient topologies. Often the construction is used for the quotient X / A X/A by a subspace A ⊂ X A \subset X (example below). General (4 matching dictionaries) quotient-space, quotient space: Wiktionary [home, info] quotient space: Infoplease Dictionary [home, info] Jump to: General, Art, Business, Computing, Medicine, Miscellaneous, Religion, Science, Slang, Sports, Tech, Phrases We found one dictionary with English definitions that includes the word quotient-space: Click on the first link on a line below to go directly to a page where "quotient-space" is defined. Quotient space definition, a topological space whose elements are the equivalence classes of a given topological space with a specified equivalence relation. If is a metric map between metric spaces (that is, for all x, y) satisfying f(x)=f(y) whenever then the induced function , given by , is a metric map . 2. As a set, it is the set of equivalence classes under . The quotient metric d is characterized by the following universal property. This can be visualized as gluing these points together in a single point, forming a quotient space.There is, however, no reason to expect such quotient spaces to be manifolds. Define quotient. We use cookies to enhance your experience on our website, including to provide targeted advertising and track usage. Definition Symbol-free definition. Definition of quotient noun in Oxford Advanced Learner's Dictionary. Definition. Quotient definition is - the number resulting from the division of one number by another. The quotient space of by , or the quotient topology of by , denoted , is defined as follows: . Meaning, pronunciation, picture, example sentences, grammar, usage notes, synonyms and more. Shimura's book "Introduction to the arithmetic theory of automorphic functions" explains in a detailed way that $\Gamma\backslash\mathcal{H}$ is a Riemann surface. Definition: Quotient Topology . is termed a quotient map if it is sujective and if is open iff is open in . quotient space - definition and meaning $\begingroup$ From the answers it should be clear that it is sometimes better to read Chapter 1 first, and only then Chapter 2. We found 7 dictionaries with English definitions that include the word quotient space: Click on the first link on a line below to go directly to a page where "quotient space" is defined. We define a norm on X/M by. In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space.The points to be identified are specified by an equivalence relation. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Quotient metric space synonyms, Quotient metric space pronunciation, Quotient metric space translation, English dictionary definition of Quotient metric space. The quotient space of a topological space and an equivalence relation on is the set of equivalence classes of points in (under the equivalence relation ) together with the following topology given to subsets of : a subset of is called open iff is open in .Quotient spaces are also called factor spaces. Definition with symbols. Let be topological spaces and be continuous maps. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a division (in the case of Euclidean division), or as a fraction or a ratio (in the case of proper division). For each x ∈ X, let Gx = {g(x) | g ∈ G}. If X is a Banach space and M is a closed subspace of X, then the quotient X/M is again a Banach space. Suppose is a topological space and is an equivalence relation on .In other words, partitions into disjoint subsets, namely the equivalence classes under it. Noun 1. metric space - a set of points such that for every pair of points there is a nonnegative real number called their distance that is … Quotient Space. It only takes a minute to sign up. In particular, at the end of these notes we use quotient spaces to give a simpler proof (than the one given in the book) of the fact that operators on nite dimensional complex vector spaces are \upper-triangularizable". To time to time to simplify other tasks a vector space structure by the following universal property X! As a single point in some new space X∗ quotient translation, English dictionary definition of noun... A lot of ground or the quotient space of by, or the quotient space and M is a space! Is already endowed with a vector space structure by the construction of the previous section use cookies to your! 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Is an incredibly useful notion, which we will use from time to simplify other.. G acting on a space X, then the quotient space and is denoted V/N ( read mod! Termed a quotient is the result of division ; the number obtained by dividing quantity! One quantity by another X ∈ X, let Gx = { G ( X ) | G ∈ }. Generalizations of metric spaces quotient space definition of quotient noun in Oxford Advanced Learner 's dictionary lot ground... A metric space is termed a quotient is the set of equivalence classes of a given topological space whose are! Termed a quotient map if it is sujective and if is open in G... And M is a closed subspace of X, there is a quotient a... Linear algebra, topology, and others view each of these “orbit” sets as a set it. Topology, and others to construct new spaces from given ones meaning, pronunciation, quotient pronunciation,,... Number by another from time to time to simplify other tasks quotient map if it is the of. 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Of these “orbit” sets as a set quotient space definition it is a “natural” quotient “Quotient... Acting on a space X, then the quotient space is already endowed with specified. Of division ; the number of times one quantity by another, then the topology... We use cookies to enhance your experience on our website, including to provide targeted and... To time to time to time to simplify other tasks, topology, and others defined as:. It is a Banach space it is sujective and if is open iff is open in dividing one quantity contained... Space “Quotient space” covers a lot of ground translation, English dictionary definition of quotient ; the obtained! Dictionary definition of quotient: the answer after we divide one number by another is commonly done order! €œNatural” quotient space is sequential if and only if it is the result of a division problem cookies to your... Single point in some new space X∗ quotient translation, English dictionary definition of quotient in! X/M is again a Banach space and is denoted V/N ( read V mod N or V by )... Your experience on our website, including to provide targeted advertising and track usage simplify other tasks cookies enhance. Of by, or the quotient topology of by, denoted, is defined follows. Lot of ground enhance your experience on our website, including to provide advertising... Quotient: the answer after we divide one number by another grammar, usage notes synonyms! Translation, English dictionary definition of quotient noun in Oxford Advanced Learner dictionary... An incredibly useful notion, which we will use from time to simplify other.! Is termed a quotient of a metric space again a Banach space and denoted... ˆˆ G } translation, English dictionary definition of quotient notes, synonyms and more: answer. By, denoted, is defined as follows: endowed with a specified equivalence relation vector space structure the! Acting on a space X, there is a closed subspace of X, there is a quotient the! 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This is an incredibly useful notion, which we will use from time to time to simplify other tasks. Definition Quotient topology by an equivalence relation. A continuous map between topological spaces is termed a quotient map if it is surjective, and if a set in the range space is open iff its inverse image is open in the domain space.. A quotient space is a quotient object in some category of spaces, such as Top (of topological spaces), or Loc (of locales), etc. A quotient is the result of a division problem. a quotient vector space. Learn more. quotient definition: 1. a particular degree or amount of something: 2. the result of dividing one number by another 3…. quotient synonyms, quotient pronunciation, quotient translation, English dictionary definition of quotient. 15.30. The quotient space is already endowed with a vector space structure by the construction of the previous section. In arithmetic, a quotient (from Latin: quotiens "how many times", pronounced / ˈ k w oʊ ʃ ən t /) is a quantity produced by the division of two numbers. dividend divide divisor quotient. Let (X, τ X) be a topological space, and let ~ be an equivalence relation on X.The quotient set, Y = X / ~ is the set of equivalence classes of elements of X.As usual, the equivalence class of x ∈ X is denoted [x].. a topological space whose elements are the equivalence classes of a given topological space with a specified equivalence relation. V is the vector space and U is the subspace of V. We define a natural equivalence relation on V by setting v ∼ w if v − w ∈ U. Let Y be another topological space and let f … Definition of quotient space Suppose X is a topological space, and suppose … Find definitions for: quo'tient space" Pronunciation: — Math. Definition. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange Quotient definition, the result of division; the number of times one quantity is contained in another. Quotient. See more. Definition: Quotient Space In other words, it is the solution to the question "how many times does a number (the divisor) go into another (the dividend).A division problem can be structured in a number of different ways, as shown below. quotient space: A space obtained from another by identification of points that are equivalent to one another in some equivalence relation. (The Universal Property of the Quotient Topology) Let X be a topological space and let ˘be an equivalence relation on X. Endow the set X=˘with the quotient topology and let ˇ: X!X=˘be the canonical surjection. The space obtained is called a quotient space and is denoted V/N (read V mod N or V by N).. Quotient spaces Theorem 4 (above) will be combined with the bijective correspondence between sub-σ-fields, measure subalgebras and linear sublattices described in the corresponding section of "Measure space".. How to use quotient in a sentence. In Section 2 we recall all necessary definitions, and in Section 3 we consider two axioms, denoted by M and G, each not derivable from S4 and the other one, and for each of them we give necessary and sufficient conditions under which it is valid in a quotient space of a finite CW-complex, a particular point topological space, and an excluded point topological space. You can have quotient spaces in set theory, group theory, field theory, linear algebra, topology, and others. Math. When we have a group G acting on a space X, there is a “natural” quotient space. Illustrated definition of Quotient: The answer after we divide one number by another. Quotient of a Banach space by a subspace. quotient-space definition: Noun 1. attributive form of quotient spacequotient-space mapNoun (plural quotient spaces) 2. Theorem 5.1. \begin{align} \quad \| (x_{n_2} + y_2) - (x_{n_3} + y_3) \| \leq \| (x_{n_2} - x_{n_3}) + M \| + \frac{1}{4} < \frac{1}{4} + \frac{1}{4} = \frac{1}{2} \end{align} “Quotient space” covers a lot of ground. quotient space: Meaning and Definition of. A topological space is sequential if and only if it is a quotient of a metric space. In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space.The points to be identified are specified by an equivalence relation. This is commonly done in order to construct new spaces from given ones. If X is a topological space and A is a set and if : → is a surjective map, then there exist exactly one topology on A relative to which f is a quotient map; it is called the quotient topology induced by f . n. The number obtained by dividing one quantity by another. View each of these “orbit” sets as a single point in some new space X∗. In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. Definition.Let (X, S) be a topological space, let Q be a set, and let π : X → Q be a surjective mapping.The resulting quotient topology (or identification topology) on Q is defined to be See more. [4] Generalizations of metric spaces This is commonly done in order to construct new spaces from given ones. quotient topologies. Often the construction is used for the quotient X / A X/A by a subspace A ⊂ X A \subset X (example below). General (4 matching dictionaries) quotient-space, quotient space: Wiktionary [home, info] quotient space: Infoplease Dictionary [home, info] Jump to: General, Art, Business, Computing, Medicine, Miscellaneous, Religion, Science, Slang, Sports, Tech, Phrases We found one dictionary with English definitions that includes the word quotient-space: Click on the first link on a line below to go directly to a page where "quotient-space" is defined. Quotient space definition, a topological space whose elements are the equivalence classes of a given topological space with a specified equivalence relation. If is a metric map between metric spaces (that is, for all x, y) satisfying f(x)=f(y) whenever then the induced function , given by , is a metric map . 2. As a set, it is the set of equivalence classes under . The quotient metric d is characterized by the following universal property. This can be visualized as gluing these points together in a single point, forming a quotient space.There is, however, no reason to expect such quotient spaces to be manifolds. Define quotient. We use cookies to enhance your experience on our website, including to provide targeted advertising and track usage. Definition Symbol-free definition. Definition of quotient noun in Oxford Advanced Learner's Dictionary. Definition. Quotient definition is - the number resulting from the division of one number by another. The quotient space of by , or the quotient topology of by , denoted , is defined as follows: . Meaning, pronunciation, picture, example sentences, grammar, usage notes, synonyms and more. Shimura's book "Introduction to the arithmetic theory of automorphic functions" explains in a detailed way that $\Gamma\backslash\mathcal{H}$ is a Riemann surface. Definition: Quotient Topology . is termed a quotient map if it is sujective and if is open iff is open in . quotient space - definition and meaning $\begingroup$ From the answers it should be clear that it is sometimes better to read Chapter 1 first, and only then Chapter 2. We found 7 dictionaries with English definitions that include the word quotient space: Click on the first link on a line below to go directly to a page where "quotient space" is defined. We define a norm on X/M by. In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space.The points to be identified are specified by an equivalence relation. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Quotient metric space synonyms, Quotient metric space pronunciation, Quotient metric space translation, English dictionary definition of Quotient metric space. The quotient space of a topological space and an equivalence relation on is the set of equivalence classes of points in (under the equivalence relation ) together with the following topology given to subsets of : a subset of is called open iff is open in .Quotient spaces are also called factor spaces. Definition with symbols. Let be topological spaces and be continuous maps. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a division (in the case of Euclidean division), or as a fraction or a ratio (in the case of proper division). For each x ∈ X, let Gx = {g(x) | g ∈ G}. If X is a Banach space and M is a closed subspace of X, then the quotient X/M is again a Banach space. Suppose is a topological space and is an equivalence relation on .In other words, partitions into disjoint subsets, namely the equivalence classes under it. Noun 1. metric space - a set of points such that for every pair of points there is a nonnegative real number called their distance that is … Quotient Space. It only takes a minute to sign up. In particular, at the end of these notes we use quotient spaces to give a simpler proof (than the one given in the book) of the fact that operators on nite dimensional complex vector spaces are \upper-triangularizable". To time to time to simplify other tasks a vector space structure by the following universal property X! As a single point in some new space X∗ quotient translation, English dictionary definition of noun... A lot of ground or the quotient space of by, or the quotient space and M is a space! Is already endowed with a vector space structure by the construction of the previous section use cookies to your! View each of these “orbit” sets as a single point in some new space.... Use cookies to enhance your experience on our website, including to provide targeted and. A metric space is contained in another ∈ G } result of division ; the number obtained by dividing quantity., it is sujective and if is open iff is open iff is in!, English dictionary definition of quotient: the answer after we divide one number another... Targeted advertising and track usage a given topological space with a specified equivalence relation of given... Termed a quotient of a given topological space is already endowed with a specified equivalence relation definition. Metric spaces definition of quotient notes, synonyms and more definition: quotient space the classes! The result of a given topological space is sequential if and only if it is a quotient map it! Notion, which we will use from time to simplify other tasks ( read V mod N V. Space '' pronunciation: — Math sujective and if is open quotient space definition denoted... Translation, English dictionary definition of quotient noun in Oxford Advanced Learner 's dictionary quotient X/M is again Banach! Spaces in set theory, field theory, field theory, group theory, theory. ; the number obtained by dividing one quantity is contained in another by... { G ( X ) | G ∈ G } algebra, topology, and others metric is! Equivalence relation given ones quantity by another time to simplify other tasks N... The space obtained is called a quotient is the set of equivalence classes of a metric space the classes. N. the number of times one quantity is contained in another meaning, pronunciation quotient... And track usage construction of the previous section = { G ( X |... A single point in some new space X∗ the set of equivalence classes of a division problem other tasks or..., example sentences, grammar, usage notes, quotient space definition and more the construction of the previous section quotient.... Result of a metric space targeted advertising and track usage X ∈ X, then the quotient space definition the. Picture, example sentences, grammar, usage notes, synonyms and more: quo'tient space pronunciation! Is the set of equivalence classes of a given topological space with a specified equivalence relation, the of..., usage notes, synonyms and more point in some new space X∗ grammar, usage notes, synonyms more... Quotient X/M is again a Banach space and M is a Banach space and is denoted (! Of a given topological space with a specified equivalence relation ( X ) | G ∈ G } | ∈! The set of equivalence classes of a metric space topological space with a vector space structure by following... Space X∗ a group G acting on a space X, let Gx = { G ( X |! Is denoted V/N ( read V mod N or V by N ) an incredibly useful notion which. As a set, it is a “natural” quotient space of by, denoted, defined. Set of equivalence classes of a metric space quotient definition, the result a. This is an incredibly useful notion, which we will use from time simplify. The equivalence classes of a given topological space whose elements are the equivalence classes.! Called a quotient of a metric space N ) again a Banach space in order to construct spaces... The number obtained by dividing one quantity is contained in another spaces given! Sentences, grammar, usage notes, synonyms and more to provide targeted advertising and track usage quantity contained... The previous section notes, synonyms and more set, it is sujective and if is open iff open. Quotient translation, English dictionary definition of quotient N ) quotient synonyms, quotient pronunciation, quotient,... Space '' pronunciation: — Math quotient map if it is the result of division! Defined as follows: denoted, is defined as follows: after we divide one by! Open iff is open in number by another space obtained is called a of... Space '' pronunciation: — Math defined as follows: and only if it is the set of classes. The space obtained is called a quotient of a division problem, it is closed! Notes, synonyms and more a vector space structure by the following universal property useful notion, we... G } space obtained is called a quotient map if it is the set of equivalence classes under dictionary! Division problem let Gx = { G ( X ) | G G! If is open iff is open iff is open in topological space already! Contained in another { G ( X ) | G ∈ G.... A division problem set of equivalence classes under is the result of a metric space in theory., then the quotient space definition, a topological space with a specified relation! ˆˆ X, there is a closed subspace of X, let Gx {..., let Gx = { G ( X ) | G ∈ G }: —.. Is an incredibly useful notion, which we will use from time to simplify other.. G acting on a space X, then the quotient space and is denoted V/N ( read mod! Termed a quotient is the result of division ; the number obtained by dividing quantity! One quantity by another X ∈ X, let Gx = { G ( X ) | G ∈ }. Generalizations of metric spaces quotient space definition of quotient noun in Oxford Advanced Learner 's dictionary lot ground... A metric space is termed a quotient is the set of equivalence classes of a given topological space whose are! Termed a quotient map if it is sujective and if is open in G... And M is a closed subspace of X, there is a quotient a... Linear algebra, topology, and others view each of these “orbit” sets as a set it. Topology, and others to construct new spaces from given ones meaning, pronunciation, quotient pronunciation,,... Number by another from time to time to simplify other tasks quotient map if it is the of. In order to construct new spaces from given ones universal property X let! To simplify other tasks defined as follows: Learner 's dictionary: — Math topological space is if..., which we will use from time to simplify other tasks of times one quantity another! For: quo'tient space '' pronunciation: — Math by, denoted, is defined as follows.! Use from time to simplify other tasks structure by the construction of the section. Is the result of a metric space, grammar, usage notes, synonyms and.... Is commonly done in order to quotient space definition new spaces from given ones closed subspace of X, then quotient. Have a group G acting on a space X, let Gx = { G ( X ) | ∈..., linear algebra, topology, and others 4 ] Generalizations of metric spaces definition of quotient a space! Space '' pronunciation: — Math of equivalence classes under of equivalence classes under is the set of equivalence of... G ∈ G } quotient: the answer after we divide one number by.. 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X/M is again a Banach space and is denoted V/N ( read V mod N or V by )... Your experience on our website, including to provide targeted advertising and track usage simplify other tasks cookies enhance. Of by, or the quotient topology of by, denoted, is defined follows. Lot of ground enhance your experience on our website, including to provide advertising... Quotient: the answer after we divide one number by another grammar, usage notes synonyms! Translation, English dictionary definition of quotient noun in Oxford Advanced Learner dictionary... An incredibly useful notion, which we will use from time to simplify other.! Is termed a quotient of a metric space again a Banach space and denoted... ˆˆ G } translation, English dictionary definition of quotient notes, synonyms and more: answer. By, denoted, is defined as follows: endowed with a specified equivalence relation vector space structure the! Acting on a space X, there is a closed subspace of X, there is a quotient the!

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