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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an embedding (or imbedding) is one instance of some
mathematical structure In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additional ...
contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is given by some
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
and structure-preserving map f:X\rightarrow Y. 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 is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, a structure-preserving map is called a
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
. The fact that a map f:X\rightarrow Y 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 numbers in the integers, the integers in the rational numbers, the rational numbers in the real numbers, and the real numbers in the complex numbers. In such cases it is common to identify the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
X with its
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
f(X) contained in Y, so that f(X)\subseteq Y.


Topology and geometry


General topology

In general topology, an embedding is a homeomorphism onto its image. More explicitly, an injective
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
map f : X \to Y between topological spaces X and Y is a topological embedding if f yields a homeomorphism between X and f(X) (where f(X) carries the subspace topology inherited from Y). Intuitively then, the embedding f : X \to Y lets us treat X as a subspace of Y. Every embedding is injective and
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
. Every map that is injective, continuous and either open or
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
is an embedding; however there are also embeddings which are neither open nor closed. The latter happens if the image f(X) is neither an open set nor a
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
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.


Related definitions

If the domain of a function f : X \to Y is a topological space then the function is said to be ' if there exists some
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
U of this point such that the restriction f\big\vert_U : U \to Y is injective. It is called ' if it is locally injective around every point of its domain. Similarly, a ' is a function for which every point in its domain has some neighborhood to which its restriction is a (topological, resp. smooth) embedding. Every injective function is locally injective but not conversely.
Local diffeomorphism In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below. Formal ...
s, local homeomorphisms, and smooth immersions are all locally injective functions that are not necessarily injective. The inverse function theorem gives a sufficient condition for a continuously differentiable function to be (among other things) locally injective. Every fiber of a locally injective function f : X \to Y is necessarily a discrete subspace of its
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
X.


Differential topology

In
differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
: Let M and N be smooth
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s and f:M\to N be a smooth map. Then f is called an immersion if its derivative is everywhere injective. An embedding, or a smooth embedding, is defined to be an 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 diffeomorphic to its image, and in particular the image of an embedding must be a submanifold. An immersion is precisely 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, RP^2 can be immersed in \mathbb^3 as is explicitly shown by
Boy's surface In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901. He discovered it on assignment from David Hilbert to prove that the projective plane ''could not'' be immersed in 3-space ...
—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 boundaries: one requires the map f: X \rightarrow Y to be such that *f(\partial X) = f(X) \cap \partial Y, and *f(X) is
transverse Transverse may refer to: *Transverse engine, an engine in which the crankshaft is oriented side-to-side relative to the wheels of the vehicle *Transverse flute, a flute that is held horizontally * Transverse force (or ''Euler force''), the tangen ...
to \partial Y in any point of f(\partial X). The first condition is equivalent to having f(\partial X) \subseteq \partial Y and f(X \setminus \partial X) \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 manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
s or more generally pseudo-Riemannian manifolds. An isometric embedding is a smooth embedding f:M\rightarrow N which preserves the (pseudo-) metric in the sense that g is equal to the pullback of h by f, i.e. g=f*h. Explicitly, for any two tangent vectors v,w\in T_x(M) we have :g(v,w)=h(df(v),df(w)). 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
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
s (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
algebraic category In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the ...
C, an embedding between two C-algebraic structures X and Y is a C-morphism that is injective.


Field theory

In field theory, an embedding of a field E in a field F is a ring homomorphism . The kernel of \sigma is an
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
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 In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the subfield \sigma(E) of F. This justifies the name ''embedding'' for an arbitrary homomorphism of fields.


Universal algebra and model theory

If \sigma is a signature and A,B are \sigma- structures (also called \sigma-algebras in universal algebra or models in
model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
), then a map h:A \to B is a \sigma-embedding
iff In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicon ...
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(f^A(a_1,\ldots,a_n))=f^B(h(a_1),\ldots,h(a_n)), * for every n-ary relation symbol R \in\sigma and a_1,\ldots,a_n \in A^n, we have A \models R(a_1,\ldots,a_n) iff B \models R(h(a_1),\ldots,h(a_n)). Here A\models R (a_1,\ldots,a_n) is a model theoretical notation equivalent to (a_1,\ldots,a_n)\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(x_1)\leq F(x_2). Injectivity of F follows quickly from this definition. In domain theory, an additional requirement is that : \forall y\in Y:\ is directed.


Metric spaces

A mapping \phi: X \to Y of metric spaces is called an ''embedding'' (with distortion C>0) if : L d_X(x, y) \leq d_Y(\phi(x), \phi(y)) \leq CLd_X(x,y) for every x,y\in X and 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 In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
(X, \, \cdot \, ) is, ''what is the maximal dimension k such that the
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
\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 is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, 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 extremal monomorphism is an embedding and embeddings are stable under pullbacks. Ideally the class of all embedded
subobject In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory,M ...
s of a given object, up to isomorphism, should also be
small Small may refer to: Science and technology * SMALL, an ALGOL-like programming language * Small (anatomy), the lumbar region of the back * ''Small'' (journal), a nano-science publication * <small>, an HTML element that defines smaller text ...
, 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 f:A\rightarrow 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 f is a morphism fg:C\rightarrow B, then g itself is a morphism. A
factorization system In mathematics, it can be shown that every function can be written as the composite of a surjective function followed by an injective function. Factorization systems are a generalization of this situation in category theory. Definition A factoriza ...
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 Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
concept, known as quotient. All the preceding properties can be dualized. An embedding can also refer to an embedding functor.


See also

* Closed immersion *
Cover Cover or covers may refer to: Packaging * Another name for a lid * Cover (philately), generic term for envelope or package * Album cover, the front of the packaging * Book cover or magazine cover ** Book design ** Back cover copy, part of co ...
* Dimension reduction * Immersion * Johnson–Lindenstrauss lemma * Submanifold * Subspace *
Universal space In mathematics, a universal space is a certain metric space that contains all metric spaces whose dimension is bounded by some fixed constant. A similar definition exists in topological dynamics. Definition Given a class \textstyle \mathcal of top ...


Notes


References

* * * * * * * * * * * * . * * .


External links

*
Embedding of manifolds
on the Manifold Atlas {{set index article Abstract algebra Category theory General topology Differential topology Functions and mappings Maps of manifolds Model theory Order theory