Partition Of Unity
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Partition Of Unity
In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ..., a partition of unity of a topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a ... ''X'' is a set ''R'' of continuous function In mathematics, a continuous function is a function (mathematics), function such that a continuous variation (that is a change without jump) of the argument of a function, argument induces a continuous variation of the Value (mathematics), value o ...s from ''X'' to the unit interval In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics a ...
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Mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus about its exact scope or . Most of mathematical activity consists of discovering and proving (by pure reasoning) properties of . These objects are either s from nature (such as s or "a "), or (in modern mathematics) abstract entities that are defined by their basic properties, called s. A consists of a succession of applications of some to already known results, including previously proved s, axioms and (in case of abstraction from nature) some basic properties that are considered as true starting points of the theory under consideration. The result of a proof is called a ''theorem''. Contrary to s, the validity of a theorem (its truth) does not rely on any but on the correctness of its reasoning (though experimentation is often useful for ...
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Category (mathematics)
In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ..., a category (sometimes called an abstract category to distinguish it from a concrete category In mathematics, a concrete category is a category (category theory), category that is equipped with a faithful functor to the category of sets (or sometimes to another category, ''see #Relative concreteness, Relative concreteness below''). This func ...) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of setsIn the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el ...
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Gluing Axiom
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ..., the gluing axiom is introduced to define what a sheaf Sheaf may refer to: * Sheaf (agriculture) A sheaf (/ʃiːf/) is a bunch of cereal-crop stems bound together after reaping, traditionally by sickle, later by scythe or, after its introduction in 1872, by a mechanical reaper-binder. Traditional ... \mathcal F on a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ... X must satisfy, given that it is a presheaf In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...
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