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Injective composition: the second function need not be injective. A function is injective (one-to-one) if each possible element of the codomain is mapped to by at most one argument. Equivalently, a function is injective if it maps distinct arguments to distinct images. An injective function is an injection. [1] The formal definition is the ...
In mathematics, an injective function (also known as injection, or one-to-one function [1]) is a function f that maps distinct elements of its domain to distinct elements of its codomain; that is, x 1 ≠ x 2 implies f(x 1) ≠ f(x 2) (equivalently by contraposition, f(x 1) = f(x 2) implies x 1 = x 2).
This and other analogous injective functions [3] from substructures are sometimes called natural injections. Given any morphism f {\displaystyle f} between objects X {\displaystyle X} and Y {\displaystyle Y} , if there is an inclusion map ι : A → X {\displaystyle \iota :A\to X} into the domain X {\displaystyle X} , then one can form the ...
Given a map :, the mapping cylinder is a space , together with a cofibration ~: and a surjective homotopy equivalence (indeed, Y is a deformation retract of ), such that the composition equals f. Thus the space Y gets replaced with a homotopy equivalent space M f {\displaystyle M_{f}} , and the map f with a lifted map f ~ {\displaystyle {\tilde ...
A faithful functor need not be injective on objects or morphisms. That is, two objects X and X′ may map to the same object in D (which is why the range of a full and faithful functor is not necessarily isomorphic to C), and two morphisms f : X → Y and f′ : X′ → Y′ (with different domains/codomains) may map to the same morphism in D.
In general topology, an embedding is a homeomorphism onto its image. [3] More explicitly, an injective continuous map : between topological spaces and is a topological embedding if yields a homeomorphism between and () (where () carries the subspace topology inherited from ).
In the category of metric spaces, Met, an injective object is an injective metric space, and the injective hull of a metric space is its tight span. In the category of T 0 spaces and continuous mappings, an injective object is always a Scott topology on a continuous lattice, and therefore it is always sober and locally compact.
In the case of maps f : U → C defined on an open subset U of the complex plane C, some authors (e.g., Freitag 2009, Definition IV.4.1) define a conformal map to be an injective map with nonzero derivative i.e., f’(z)≠ 0 for every z in U. According to this definition, a map f : U → C is conformal if and only if f: U → f(U) is ...