Quotient. If X is a topological space and A is a set and if : â 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 . Definition Quotient topology by an equivalence relation. The quotient metric d is characterized by the following universal property. 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. a topological space whose elements are the equivalence classes of a given topological space with a specified equivalence relation. quotient synonyms, quotient pronunciation, quotient translation, English dictionary definition of quotient. 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. It only takes a minute to sign up. Often the construction is used for the quotient X / A X/A by a subspace A â X A \subset X (example below). 15.30. Definition of quotient space Suppose X is a topological space, and suppose â¦ n. The number obtained by dividing one quantity by another. 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. This is commonly done in order to construct new spaces from given ones. âQuotient spaceâ covers a lot of ground. Definition Symbol-free definition. We use cookies to enhance your experience on our website, including to provide targeted advertising and track usage. 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". Definition of quotient noun in Oxford Advanced Learner's Dictionary. Quotient space definition, a topological space whose elements are the equivalence classes of a given topological space with a specified equivalence relation. Learn more. When we have a group G acting on a space X, there is a ânaturalâ quotient space. How to use quotient in a sentence. In arithmetic, a quotient (from Latin: quotiens "how many times", pronounced / Ë k w oÊ Ê Én t /) is a quantity produced by the division of two numbers. Definition with symbols. General (4 matching dictionaries) quotient-space, quotient space: Wiktionary [home, info] quotient space: Infoplease Dictionary [home, info] (The Universal Property of the Quotient Topology) Let X be a topological space and let Ëbe an equivalence relation on X. Endow the set X=Ëwith the quotient topology and let Ë: X!X=Ëbe the canonical surjection. 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. 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.. quotient definition: 1. a particular degree or amount of something: 2. the result of dividing one number by another 3â¦. As a set, it is the set of equivalence classes under . quotient space: A space obtained from another by identification of points that are equivalent to one another in some equivalence relation. Find definitions for: quo'tient space" Pronunciation: â Math. Quotient spaces Theorem 4 (above) will be combined with the bijective correspondence between sub-Ï-fields, measure subalgebras and linear sublattices described in the corresponding section of "Measure 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. 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. Definition: Quotient Topology . Quotient definition is - the number resulting from the division of one number by another. 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. Let be topological spaces and be continuous maps. V is the vector space and U is the subspace of V. We define a natural equivalence relation on V by setting v â¼ w if v â w â U. 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 â¦ A quotient is the result of a division problem. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. A topological space is sequential if and only if it is a quotient of a metric space. Quotient metric space synonyms, Quotient metric space pronunciation, Quotient metric space translation, English dictionary definition of Quotient metric space. Quotient of a Banach space by a subspace. You can have quotient spaces in set theory, group theory, field theory, linear algebra, topology, and others. Define quotient. Quotient definition, the result of division; the number of times one quantity is contained in another. The space obtained is called a quotient space and is denoted V/N (read V mod N or V by N).. Definition.Let (X, S) be a topological space, let Q be a set, and let Ï : X â Q be a surjective mapping.The resulting quotient topology (or identification topology) on Q is defined to be quotient topologies. a quotient vector space. View each of these âorbitâ sets as a single point in some new space Xâ. The quotient space is already endowed with a vector space structure by the construction of the previous section. 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 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 . is termed a quotient map if it is sujective and if is open iff is open in . $\begingroup$ From the answers it should be clear that it is sometimes better to read Chapter 1 first, and only then Chapter 2. Let Y be another topological space and let f â¦ Definition: Quotient Space In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. Math. 2. This is commonly done in order to construct new spaces from given ones. We define a norm on X/M by. quotient space: Meaning and Definition of. If X is a Banach space and M is a closed subspace of X, then the quotient X/M is again a Banach space. Definition. See more. \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} See more. This is an incredibly useful notion, which we will use from time to time to simplify other tasks. quotient-space definition: Noun 1. attributive form of quotient spacequotient-space mapNoun (plural quotient spaces) 2. Theorem 5.1. dividend divide divisor quotient. Quotient Space. Definition. Meaning, pronunciation, picture, example sentences, grammar, usage notes, synonyms and more. 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). 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. quotient space - definition and meaning The quotient space of by , or the quotient topology of by , denoted , is defined as follows: . For each x â X, let Gx = {g(x) | g â G}. 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. Let (X, Ï 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 â X is denoted [x].. Illustrated definition of Quotient: The answer after we divide one number by another. 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