TheInfoList

In
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, an embedding (or imbedding) is one instance of some
mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
contained within another instance, such as a group that is a
subgroup In group theory, a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathema ...
. When some object ''X'' is said to be embedded in another object ''Y'', the embedding is given by some
injective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

and structure-preserving map . The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which ''X'' and ''Y'' are instances. In the terminology of
category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled directed edges are cal ...
, a structure-preserving map is called a morphism. The fact that a map is an embedding is often indicated by the use of a "hooked arrow" (); thus: $f : X \hookrightarrow Y.$ (On the other hand, this notation is sometimes reserved for inclusion maps.) Given ''X'' and ''Y'', several different embeddings of ''X'' in ''Y'' may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the
natural number In mathematics, the natural numbers are those used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, words coll ...
s in the
integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, and −2048 are integers, while 9 ...
s, the integers in the rational numbers, the rational numbers in the
real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R\$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...
s, and the real numbers in the
complex number In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...
s. In such cases it is common to identify the domain ''X'' with its
image An SAR radar imaging, radar image acquired by the SIR-C/X-SAR radar on board the Space Shuttle Endeavour shows the Teide volcano. The city of Santa Cruz de Tenerife is visible as the purple and white area on the lower right edge of the island ...
''f''(''X'') contained in ''Y'', so that .

# Topology and geometry

## General topology

In
general topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics, general topology is the branch of topology that deals with the basic Set theory, set-theoretic definitions and constructions used in topology. It is t ...
, an embedding is a homeomorphism onto its image. More explicitly, an injective continuous map $f : X \to Y$ between
topological space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s $X$ and $Y$ is a topological embedding if $f$ yields a homeomorphism between $X$ and $f\left(X\right)$ (where $f\left(X\right)$ carries the subspace topology inherited from $Y$). Intuitively then, the embedding $f : X \to Y$ lets us treat $X$ as a subspace topology, subspace of $Y$. Every embedding is injective and continuous. Every map that is injective, continuous and either open map, open or closed map, closed is an embedding; however there are also embeddings which are neither open nor closed. The latter happens if the image $f\left(X\right)$ is neither an open set nor a closed set in $Y$. For a given space $Y$, the existence of an embedding $X \to Y$ is a topological invariant of $X$. This allows two spaces to be distinguished if one is able to be embedded in a space while the other is not.

## Differential topology

In differential topology: Let $M$ and $N$ be smooth manifolds and $f:M\to N$ be a smooth map. Then $f$ is called an immersion (mathematics), immersion if its pushforward (differential), derivative is everywhere injective. An embedding, or a smooth embedding, is defined to be an injective immersion which is an embedding in the topological sense mentioned above (i.e. homeomorphism onto its image). In other words, the domain of an embedding is diffeomorphism, diffeomorphic to its image, and in particular the image of an embedding must be a submanifold. An immersion is a local embedding (i.e. for any point $x\in M$ there is a neighborhood $x\in U\subset M$ such that $f:U\to N$ is an embedding.) When the domain manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion. An important case is $N = \mathbb^n$. The interest here is in how large $n$ must be for an embedding, in terms of the dimension $m$ of $M$. The Whitney embedding theorem states that $n = 2m$ is enough, and is the best possible linear bound. For example, the real projective space RP''m'' of dimension $m$, where $m$ is a power of two, requires $n = 2m$ for an embedding. However, this does not apply to immersions; for instance, RP2 can be immersed in $\mathbb^3$ as is explicitly shown by Boy's surface—which has self-intersections. The Roman surface fails to be an immersion as it contains cross-caps. An embedding is proper if it behaves well with respect to Topological manifold#Manifolds with boundary, boundaries: one requires the map $f: X \rightarrow Y$ to be such that *$f\left(\partial X\right) = f\left(X\right) \cap \partial Y$, and *$f\left(X\right)$ is Transversality (mathematics), transverse to $\partial Y$ in any point of $f\left(\partial X\right)$. The first condition is equivalent to having $f\left(\partial X\right) \subseteq \partial Y$ and $f\left(X \setminus \partial X\right) \subseteq Y \setminus \partial Y$. The second condition, roughly speaking, says that ''f''(''X'') is not tangent to the boundary of ''Y''.

## Riemannian and pseudo-Riemannian geometry

In Riemannian geometry and pseudo-Riemannian geometry: Let (''M'', ''g'') and (''N'', ''h'') be Riemannian manifolds or more generally pseudo-Riemannian manifolds. An isometric embedding is a smooth embedding ''f'' : ''M'' → ''N'' which preserves the (pseudo-)Riemannian metric, metric in the sense that ''g'' is equal to the pullback (differential geometry), pullback of ''h'' by ''f'', i.e. ''g'' = ''f''*''h''. Explicitly, for any two tangent vectors $v,w\in T_x\left(M\right)$ we have :$g\left(v,w\right)=h\left(df\left(v\right),df\left(w\right)\right).$ Analogously, isometric immersion is an immersion between (pseudo)-Riemannian manifolds which preserves the (pseudo)-Riemannian metrics. Equivalently, in Riemannian geometry, an isometric embedding (immersion) is a smooth embedding (immersion) which preserves length of curves (cf. Nash embedding theorem).Nash J., ''The embedding problem for Riemannian manifolds,'' Ann. of Math. (2), 63 (1956), 20–63.

# Algebra

In general, for an Variety (universal algebra), algebraic category ''C'', an embedding between two ''C''-algebraic structures ''X'' and ''Y'' is a ''C''-morphism that is injective.

## Field theory

In field theory (mathematics), field theory, an embedding of a field (mathematics), field ''E'' in a field ''F'' is a ring homomorphism . The Kernel (algebra), kernel of ''σ'' is an ideal (ring theory), ideal of ''E'' which cannot be the whole field ''E'', because of the condition . Furthermore, it is a well-known property of fields that their only ideals are the zero ideal and the whole field itself. Therefore, the kernel is 0, so any embedding of fields is a monomorphism. Hence, ''E'' is isomorphic to the Field extension, subfield ''σ''(''E'') of ''F''. This justifies the name ''embedding'' for an arbitrary homomorphism of fields.

## Universal algebra and model theory

If σ is a signature (logic), signature and $A,B$ are σ-structure (mathematical logic), structures (also called σ-algebras in universal algebra or models in model theory), then a map $h:A \to B$ is a σ-embedding iff all of the following hold: * $h$ is injective, * for every $n$-ary function symbol $f \in\sigma$ and $a_1,\ldots,a_n \in A^n,$ we have $h\left(f^A\left(a_1,\ldots,a_n\right)\right)=f^B\left(h\left(a_1\right),\ldots,h\left(a_n\right)\right)$, * for every $n$-ary relation symbol $R \in\sigma$ and $a_1,\ldots,a_n \in A^n,$ we have $A \models R\left(a_1,\ldots,a_n\right)$ iff $B \models R\left(h\left(a_1\right),\ldots,h\left(a_n\right)\right).$ Here $A\models R \left(a_1,\ldots,a_n\right)$ is a model theoretical notation equivalent to $\left(a_1,\ldots,a_n\right)\in R^A$. In model theory there is also a stronger notion of elementary embedding.

# Order theory and domain theory

In order theory, an embedding of partially ordered sets is a function ''F'' between partially ordered sets ''X'' and ''Y'' such that :$\forall x_1,x_2\in X: x_1\leq x_2 \iff F\left(x_1\right)\leq F\left(x_2\right).$ Injectivity of ''F'' follows quickly from this definition. In domain theory, an additional requirement is that :$\forall y\in Y:\$ is Directed set, directed.

# Metric spaces

A mapping $\phi: X \to Y$ of metric spaces is called an ''embedding'' (with stretch factor, distortion $C>0$) if :$L d_X\left(x, y\right) \leq d_Y\left(\phi\left(x\right), \phi\left(y\right)\right) \leq CLd_X\left(x,y\right)$ for some constant $L>0$.

## Normed spaces

An important special case is that of normed spaces; in this case it is natural to consider linear embeddings. One of the basic questions that can be asked about a finite-dimensional normed space $\left(X, \, \cdot \, \right)$ is, ''what is the maximal dimension $k$ such that the Hilbert space $\ell_2^k$ can be linearly embedded into $X$ with constant distortion?'' The answer is given by Dvoretzky's theorem.

# Category theory

In
category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled directed edges are cal ...
, there is no satisfactory and generally accepted definition of embeddings that is applicable in all categories. One would expect that all isomorphisms and all compositions of embeddings are embeddings, and that all embeddings are monomorphisms. Other typical requirements are: any monomorphism#Related concepts, extremal monomorphism is an embedding and embeddings are stable under Pullback (category theory), pullbacks. Ideally the class of all embedded subobjects of a given object, up to isomorphism, should also be small class, small, and thus an ordered set. In this case, the category is said to be well powered with respect to the class of embeddings. This allows defining new local structures in the category (such as a closure operator). In a concrete category, an embedding is a morphism ''ƒ'': ''A'' → ''B'' which is an injective function from the underlying set of ''A'' to the underlying set of ''B'' and is also an initial morphism in the following sense: If ''g'' is a function from the underlying set of an object ''C'' to the underlying set of ''A'', and if its composition with ''ƒ'' is a morphism ''ƒg'': ''C'' → ''B'', then ''g'' itself is a morphism. A factorization system for a category also gives rise to a notion of embedding. If (''E'', ''M'') is a factorization system, then the morphisms in ''M'' may be regarded as the embeddings, especially when the category is well powered with respect to ''M''. Concrete theories often have a factorization system in which ''M'' consists of the embeddings in the previous sense. This is the case of the majority of the examples given in this article. As usual in category theory, there is a dual (category theory), dual concept, known as quotient. All the preceding properties can be dualized. An embedding can also refer to an Subcategory#Embeddings, embedding functor.

*Closed immersion *Cover (algebra), Cover *Dimension reduction *Immersion (mathematics), Immersion *Johnson–Lindenstrauss lemma *Submanifold *Subspace (topology), Subspace *Universal spaces in the topology and topological dynamics, Universal space

# References

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