Search results
Results from the WOW.Com Content Network
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 ...
A subgroup is a retract if and only if it has a normal complement. [4] The normal complement, specifically, is the kernel of the retraction. Every direct factor is a retract. [1] Conversely, any retract which is a normal subgroup is a direct factor. [5] Every retract has the congruence extension property.
The concept of a retraction in category theory comes from the essentially similar notion of a retraction in topology: : where is a subspace of is a retraction in the topological sense, if it's a retraction of the inclusion map : in the category theory sense.
A retract A of a space X with the fixed-point property also has the fixed-point property. This is because if r : X → A {\displaystyle r:X\to A} is a retraction and f : A → A {\displaystyle f:A\to A} is any continuous function, then the composition i ∘ f ∘ r : X → X {\displaystyle i\circ f\circ r:X\to X} (where i : A → X ...
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:
Retracted (phonetics), a sound pronounced to the back of the vocal tract, in linguistics; Retracted tongue root, a position of the tongue during the pronunciation of a vowel, in phonetics; Sternal retraction, a symptom of respiratory distress in humans; Retraction (kinesiology), an anatomical term of motion
A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...
An example is a neighborhood deformation retract; that is, ... by definition, ... Differential Forms in Algebraic Topology. Springer.