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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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moore Space (algebraic Topology)
In algebraic topology, a branch of mathematics, Moore space is the name given to a particular type of topological space that is the homology analogue of the Eilenberg–Maclane spaces of homotopy theory, in the sense that it has only one nonzero homology (rather than homotopy) group. Formal definition Given an abelian group ''G'' and an integer ''n'' ≥ 1, let ''X'' be a CW complex such that :H_n(X) \cong G and :\tilde_i(X) \cong 0 for ''i'' ≠ ''n'', where H_n(X) denotes the ''n''-th singular homology group of ''X'' and \tilde_i(X) is the ''i''-th reduced homology group. Then ''X'' is said to be a Moore space. Also, ''X'' is by definition simply-connected if ''n''>1. Examples *S^n is a Moore space of \mathbb for n\geq 1. *\mathbb^2 is a Moore space of \mathbb/2\mathbb for n=1. See also * Eilenberg–MacLane space, the homotopy analog. * Homology sphere Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
