Editing Embedding (section)

{{Redirect|Isometric embedding|related concepts for [[metric space]]s|isometry}}
{{For|embeddings of graphs in two-dimensional manifolds|graph embedding}}
{{Other uses}}
{{Short description|Inclusion of one mathematical structure in another, preserving properties of interest}}
In [[mathematics]], an '''embedding''' (or '''imbedding'''<ref>{{harvnb|Spivak|1999|page=49}} suggests that "the English" (i.e. the British) use "embedding" instead of "imbedding".</ref>) is one instance of some [[mathematical structure]] contained within another instance, such as a [[group (mathematics)|group]] that is a [[subgroup]].

When some object <math>X</math> is said to be embedded in another object <math>Y</math>, the embedding is given by some [[Injective function|injective]] and structure-preserving map <math>f:X\rightarrow Y</math>. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which <math>X</math> and <math>Y</math> are instances. In the terminology of [[category theory]], a structure-preserving map is called a [[morphism]].

The fact that a map <math>f:X\rightarrow Y</math> is an embedding is often indicated by the use of a "hooked arrow" ({{unichar|21AA|RIGHTWARDS ARROW WITH HOOK|ulink=Unicode}});<ref name="Unicode Arrows">{{cite web| title = Arrows – Unicode| url = https://www.unicode.org/charts/PDF/U2190.pdf| access-date = 2017-02-07}}</ref> thus: <math> f : X \hookrightarrow Y.</math> (On the other hand, this notation is sometimes reserved for [[inclusion map]]s.)

Given <math>X</math> and <math>Y</math>, several different embeddings of <math>X</math> in <math>Y</math> may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the [[natural number]]s in the [[integer]]s, the integers in the [[rational number]]s, the rational numbers in the [[real number]]s, and the real numbers in the [[complex number]]s. In such cases it is common to identify the [[Domain of a function|domain]] <math>X</math> with its [[image (mathematics)|image]] <math>f(X)</math> contained in <math>Y</math>, so that <math>X\subseteq Y</math>.