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Cylinder Set
In mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ-algebra. General definition Given a collection S of sets, consider the Cartesian product X = \prod_ Y of all sets in the collection. The canonical projection corresponding to some Y\in S is the function p_ : X \to Y that maps every element of the product to its Y component. A cylinder set is a preimage of a canonical projection or finite intersection of such preimages. Explicitly, it is a set of the form, \bigcap_^n p_^ \left(A_i\right) = \left\ for any choice of n, finite sequence of sets Y_1,...Y_n\in S and subsets A_ \subseteq Y_i for 1 \leq i \leq n. Here x_Y\in Y denotes the Y component of x\in X. Then, when all sets in S are topological spaces, the product topology is generated by cylinder sets corresponding to the components' open sets. That is cylinders of the form \bigcap_^n p_^ \left(U_i\right) where for each i, U_i is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
