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In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. [1] The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of continuously shrinking a space into a ...
In mathematics, in the field of group theory, ... The normal complement, specifically, is the kernel of the retraction. Every direct factor is a retract. [1]
In mathematics, Sharafutdinov's retraction is a construction that gives a retraction of an open non-negatively curved Riemannian manifold onto its soul. It was first used by Sharafutdinov to show that any two souls of a complete Riemannian manifold with non-negative sectional curvature are isometric. [1]
In mathematics, a projection is an idempotent mapping of a set (or other mathematical structure) into a subset (or sub-structure). In this case, idempotent means that projecting twice is the same as projecting once. The restriction to a subspace of a projection is also called a projection, even if the idempotence property is lost. An everyday ...
In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism.
Retraction in academic publishing, withdrawals of previously published academic journal articles; Mathematics. Retraction (category theory) Retract (group theory)
a retraction if a right inverse of f exists, i.e. if there exists a morphism g : b → a with f ∘ g = 1 b. a section if a left inverse of f exists, i.e. if there exists a morphism g : b → a with g ∘ f = 1 a. Every retraction is an epimorphism, and every section is a monomorphism. Furthermore, the following three statements are equivalent:
Among hundreds of fixed-point theorems, [1] Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem , the hairy ball theorem , the invariance of dimension ...