1.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3

2.
Subspace topology
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In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology. Given a topological space and a subset S of X, the topology on S is defined by τ S =. That is, a subset of S is open in the subspace topology if, if S is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of. Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated, alternatively we can define the subspace topology for a subset S of X as the coarsest topology for which the inclusion map ι, S ↪ X is continuous. More generally, suppose ι is an injection from a set S to a topological space X, then the subspace topology on S is defined as the coarsest topology for which ι is continuous. The open sets in topology are precisely the ones of the form ι −1 for U open in X. S is then homeomorphic to its image in X and ι is called a topological embedding. A subspace S is called a subspace if the injection ι is an open map. Likewise it is called a subspace if the injection ι is a closed map. The distinction between a set and a space is often blurred notationally, for convenience, which can be a source of confusion when one first encounters these definitions. In the following, R represents the numbers with their usual topology. The subspace topology of the numbers, as a subspace of R, is the discrete topology. The rational numbers Q considered as a subspace of R do not have the discrete topology. If a and b are rational, then the intervals and are open and closed. The set as a subspace of R is both open and closed, whereas as a subset of R it is only closed, as a subspace of R, ∪ is composed of two disjoint open subsets, and is therefore a disconnected space. Let S = [0, 1) be a subspace of the real line R, then [0, 1/2) is open in S but not in R. Likewise [½, 1) is closed in S but not in R. S is both open and closed as a subset of itself but not as a subset of R, the subspace topology has the following characteristic property. Let Y be a subspace of X and let i, Y → X be the inclusion map, then for any topological space Z a map f, Z → Y is continuous if and only if the composite map i ∘ f is continuous. This property is characteristic in the sense that it can be used to define the topology on Y

3.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space

4.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

5.
Topology
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, important topological properties include connectedness and compactness. Topology developed as a field of study out of geometry and set theory, through analysis of such as space, dimension. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs, Leonhard Eulers Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the fields first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, by the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to geometry and together they make up the geometric theory of differentiable manifolds. Geometric topology primarily studies manifolds and their embeddings in other manifolds, a particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots, Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler and his 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron and this led to his polyhedron formula, V − E + F =2. Some authorities regard this analysis as the first theorem, signalling the birth of topology, further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term Topologie in Vorstudien zur Topologie, written in his native German, in 1847, the term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Their work was corrected, consolidated and greatly extended by Henri Poincaré, in 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a case of a general topological space. In 1914, Felix Hausdorff coined the term topological space and gave the definition for what is now called a Hausdorff space, currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski

6.
Continuous function (topology)
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In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function, a continuous function with a continuous inverse function is called a homeomorphism. Continuity of functions is one of the concepts of topology. The introductory portion of this focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the general case of functions between two metric spaces. In order theory, especially in theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article, as an example, consider the function h, which describes the height of a growing flower at time t. By contrast, if M denotes the amount of money in an account at time t, then the function jumps at each point in time when money is deposited or withdrawn. A form of the definition of continuity was first given by Bernard Bolzano in 1817. Cauchy defined infinitely small quantities in terms of quantities. The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s but the work wasnt published until the 1930s, all three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of continuity in 1872. This is not a definition of continuity since the function f =1 x is continuous on its whole domain of R ∖ A function is continuous at a point if it does not have a hole or jump. A “hole” or “jump” in the graph of a function if the value of the function at a point c differs from its limiting value along points that are nearby. Such a point is called a discontinuity, a function is then continuous if it has no holes or jumps, that is, if it is continuous at every point of its domain. Otherwise, a function is discontinuous, at the points where the value of the function differs from its limiting value, there are several ways to make this definition mathematically rigorous. These definitions are equivalent to one another, so the most convenient definition can be used to determine whether a function is continuous or not. In the definitions below, f, I → R. is a function defined on a subset I of the set R of real numbers and this subset I is referred to as the domain of f

7.
Initial topology
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In general topology and related areas of mathematics, the initial topology on a set X, with respect to a family of functions on X, is the coarsest topology on X that makes those functions continuous. The subspace topology and product topology constructions are both special cases of initial topologies, indeed, the initial topology construction can be viewed as a generalization of these. The dual construction is called the final topology, explicitly, the initial topology may be described as the topology generated by sets of the form f i −1, where U is an open set in Y i. The sets f i −1 are often called cylinder sets, if I contains exactly one element, all the open sets of are cylinder sets. Several topological constructions can be regarded as special cases of the initial topology, the subspace topology is the initial topology on the subspace with respect to the inclusion map. The product topology is the initial topology with respect to the family of projection maps, the inverse limit of any inverse system of spaces and continuous maps is the set-theoretic inverse limit together with the initial topology determined by the canonical morphisms. The weak topology on a convex space is the initial topology with respect to the continuous linear forms of its dual space. Given a family of topologies on a fixed set X the initial topology on X with respect to the functions idi and that is, the initial topology τ is the topology generated by the union of the topologies. A topological space is regular if and only if it has the initial topology with respect to its family of real-valued continuous functions. Every topological space X has the initial topology with respect to the family of functions from X to the Sierpiński space. The initial topology on X can be characterized by the characteristic property, A function g from some space Z to X is continuous if. Note that, despite looking quite similar, this is not a universal property, a categorical description is given below. By the universal property of the topology, we know that any family of continuous maps fi. This map is known as the evaluation map, a family of maps is said to separate points in X if for all x ≠ y in X there exists some i such that fi ≠ fi. Clearly, the family separates points if and only if the evaluation map f is injective. The evaluation map f will be an embedding if and only if X has the initial topology determined by the maps. If a space X comes equipped with a topology, it is useful to know whether or not the topology on X is the initial topology induced by some family of maps on X. This section gives a sufficient condition, a family of continuous maps separates points from closed sets if and only if the cylinder sets f i −1, for U open in Yi, form a base for the topology on X

8.
Final topology
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In general topology and related areas of mathematics, the final topology on a set X, with respect to a family of functions into X, is the finest topology on X which makes those functions continuous. The dual notion is the initial topology, given a set X and a family of topological spaces Y i with functions f i, Y i → X the final topology τ on X is the finest topology such that each f i, Y i → is continuous. Explicitly, the topology may be described as follows, a subset U of X is open if. The quotient topology is the topology on the quotient space with respect to the quotient map. The disjoint union is the final topology with respect to the family of canonical injections, more generally, a topological space is coherent with a family of subspaces if it has the final topology coinduced by the inclusion maps. The direct limit of any system of spaces and continuous maps is the set-theoretic direct limit together with the final topology determined by the canonical morphisms. Given a family of topologies on a fixed set X the final topology on X with respect to the functions idX and that is, the final topology τ is the intersection of the topologies. The etale space of a sheaf is topologized by a final topology, a subset of X is closed/open if and only if its preimage under fi is closed/open in Y i for each i ∈ I. The final topology on X can be characterized by the universal property. In the language of category theory, the topology construction can be described as follows. Let Y be a functor from a discrete category J to the category of topological spaces Top which selects the spaces Yi for i in J, let Δ be the diagonal functor from Top to the functor category TopJ. The comma category is then the category of cones from Y, i. e. objects in are pairs where fi, Yi → X is a family of continuous maps to X. If U is the functor from Top to Set and Δ′ is the diagonal functor from Set to SetJ then the comma category is the category of all cones from UY. The final topology construction can then be described as a functor from to and this functor is left adjoint to the corresponding forgetful functor

9.
Subset
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In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained inside B, that is, all elements of A are also elements of B. The relationship of one set being a subset of another is called inclusion or sometimes containment, the subset relation defines a partial order on sets. The algebra of subsets forms a Boolean algebra in which the relation is called inclusion. For any set S, the inclusion relation ⊆ is an order on the set P of all subsets of S defined by A ≤ B ⟺ A ⊆ B. We may also partially order P by reverse set inclusion by defining A ≤ B ⟺ B ⊆ A, when quantified, A ⊆ B is represented as, ∀x. So for example, for authors, it is true of every set A that A ⊂ A. Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively and this usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may or may not equal y, but if x < y, then x definitely does not equal y, and is less than y. Similarly, using the convention that ⊂ is proper subset, if A ⊆ B, then A may or may not equal B, the set A = is a proper subset of B =, thus both expressions A ⊆ B and A ⊊ B are true. The set D = is a subset of E =, thus D ⊆ E is true, any set is a subset of itself, but not a proper subset. The empty set, denoted by ∅, is also a subset of any given set X and it is also always a proper subset of any set except itself. These are two examples in both the subset and the whole set are infinite, and the subset has the same cardinality as the whole. The set of numbers is a proper subset of the set of real numbers. In this example, both sets are infinite but the set has a larger cardinality than the former set. Another example in an Euler diagram, Inclusion is the partial order in the sense that every partially ordered set is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identified with the set of all ordinals less than or equal to n, then a ≤ b if and only if ⊆. For the power set P of a set S, the partial order is the Cartesian product of k = |S| copies of the partial order on for which 0 <1. This can be illustrated by enumerating S = and associating with each subset T ⊆ S the k-tuple from k of which the ith coordinate is 1 if and only if si is a member of T

10.
Inclusion map
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In mathematics, if A is a subset of B, then the inclusion map is the function ι that sends each element, x, of A to x, treated as an element of B, ι, A → B, ι = x. A hooked arrow ↪ is sometimes used in place of the function arrow above to denote an inclusion map and this and other analogous injective functions from substructures are sometimes called natural injections. Given any morphism f between objects X and Y, if there is a map into the domain ι, A → X. In many instances, one can construct a canonical inclusion into the codomain R→Y known as the range of f. Inclusion maps tend to be homomorphisms of algebraic structures, thus, more precisely, given a sub-structure closed under some operations, the inclusion map will be an embedding for tautological reasons. For example, for a binary operation ⋆, to require that ι = ι ⋆ ι is simply to say that ⋆ is consistently computed in the sub-structure and the large structure. The case of an operation is similar, but one should also look at nullary operations. Here the point is that closure means such constants must already be given in the substructure, inclusion maps are seen in algebraic topology where if A is a strong deformation retract of X, the inclusion map yields an isomorphism between all homotopy groups. Inclusion maps in geometry come in different kinds, for example embeddings of submanifolds, contravariant objects such as differential forms restrict to submanifolds, giving a mapping in the other direction. Another example, more sophisticated, is that of affine schemes, for which the inclusions Spec → Spec and Spec → Spec may be different morphisms, fundamental Concepts of Algebra, Academic Press, New York, ISBN 0-12-172050-0. Mac Lane, S. Birkhoff, G. Algebra, AMS Chelsea Publishing, Providence, Rhode Island, ISBN 0-8218-1646-2

11.
Quotient space (topology)
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In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or gluing together certain points of a given topological space. The points to be identified are specified by an equivalence relation and this is commonly done in order to construct new spaces from given ones. The quotient topology consists of all sets with an open preimage under the projection map that maps each element to its equivalence class. The quotient topology is the topology on the quotient space with respect to the map q. A map f, X → Y is a quotient map if it is surjective, equivalently, f is a quotient map if it is onto and Y is equipped with the final topology with respect to f. Given an equivalence relation ∼ on X, the map q, X → X / ∼ is a quotient map. Topologists talk of gluing points together, consider the unit square I2 = × and the equivalence relation ~ generated by the requirement that all boundary points be equivalent, thus identifying all boundary points to a single equivalence class. Then I2/~ is homeomorphic to the unit sphere S2, more generally, suppose X is a space and A is a subspace of X. One can identify all points in A to an equivalence class. The resulting quotient space is denoted X/A, the 2-sphere is then homeomorphic to the unit disc with its boundary identified to a single point, D2 / ∂ D2. Consider the set X = R of all real numbers with the ordinary topology, then the quotient space X/~ is homeomorphic to the unit circle S1 via the homeomorphism which sends the equivalence class of x to exp. A generalization of the example is the following, Suppose a topological group G acts continuously on a space X. One can form a relation on X by saying points are equivalent if. The quotient space under this relation is called the orbit space, in the previous example G = Z acts on R by translation. The orbit space R/Z is homeomorphic to S1, note, The notation R/Z is somewhat ambiguous. If Z is understood to be a group acting on R then the quotient is the circle, however, if Z is thought of as a subspace of R, then the quotient is a countably infinite bouquet of circles joined at a single point. We say that g descends to the quotient, the continuous maps defined on X/~ are therefore precisely those maps which arise from continuous maps defined on X that respect the equivalence relation. This criterion is used when studying quotient spaces

12.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker