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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).
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 ...
Inclusion maps are seen in algebraic topology where if is a strong deformation retract of , the inclusion map yields an isomorphism between all homotopy groups (that is, it is a homotopy equivalence). Inclusion maps in geometry come in different kinds: for example embeddings of submanifolds.
Functions which satisfy property (4) are said to be "one-to-one functions" and are called injections (or injective functions). [2] With this terminology, a bijection is a function which is both a surjection and an injection, or using other words, a bijection is a function which is both "one-to-one" and "onto". [3]
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 ...
For example, in the category Div of divisible (abelian) groups and group homomorphisms between them there are monomorphisms that are not injective: consider, for example, the quotient map q : Q → Q/Z, where Q is the rationals under addition, Z the integers (also considered a group under addition), and Q/Z is the corresponding quotient group ...
For example, an injective continuous map is a monomorphism in the category of topological spaces. For proving that, conversely, a left cancelable homomorphism is injective, it is useful to consider a free object on .
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 ).