\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 definition: 1. a particular degree or amount of something: 2. the result of dividing one number by another 3…. 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). General (4 matching dictionaries) quotient-space, quotient space: Wiktionary [home, info] quotient space: Infoplease Dictionary [home, info] 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 . A quotient is the result of a division problem. 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 topological space is sequential if and only if it is a quotient of a metric space. 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. Definition Quotient topology by an equivalence relation. 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. Definition of quotient noun in Oxford Advanced Learner's Dictionary. a topological space whose elements are the equivalence classes of a given topological space with a specified equivalence relation. “Quotient space” covers a lot of ground. Let be topological spaces and be continuous maps. a quotient vector space. quotient space - definition and meaning 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. (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. 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. Definition Symbol-free definition. 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: Quotient Space 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].. quotient space: A space obtained from another by identification of points that are equivalent to one another in some equivalence relation. 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. Quotient space definition, a topological space whose elements are the equivalence classes of a given topological space with a specified equivalence relation. quotient topologies. 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 Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. 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. See more. This is commonly done in order to construct new spaces from given ones. 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. 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".. Define quotient. Often the construction is used for the quotient X / A X/A by a subspace A ⊂ X A \subset X (example below). The quotient space is already endowed with a vector space structure by the construction of the previous section. 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 … 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". Meaning, pronunciation, picture, example sentences, grammar, usage notes, synonyms and more. quotient synonyms, quotient pronunciation, quotient translation, English dictionary definition of quotient. Quotient of a Banach space by a subspace. $\begingroup$ From the answers it should be clear that it is sometimes better to read Chapter 1 first, and only then Chapter 2. Definition: Quotient Topology . Definition with symbols. Quotient definition is - the number resulting from the division of one number by another. 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. 2. is termed a quotient map if it is sujective and if is open iff is open in . Math. It only takes a minute to sign up. Illustrated definition of Quotient: The answer after we divide one number by another. Quotient definition, the result of division; the number of times one quantity is contained in another. 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. The quotient space of by , or the quotient topology of by , denoted , is defined as follows: . You can have quotient spaces in set theory, group theory, field theory, linear algebra, topology, and others. In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. See more. Suppose is a topological space and is an equivalence relation on .In other words, partitions into disjoint subsets, namely the equivalence classes under it. This is an incredibly useful notion, which we will use from time to time to simplify other tasks. The quotient metric d is characterized by the following universal property. dividend divide divisor quotient. 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