Embedding
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- For other uses of this term, see Embedded (disambiguation).
In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup.
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Topology/Geometry
General topology
In general topology, an embedding is a homeomorphism onto its image. More explicitly, a map f : X → Y between topological spaces X and Y is an 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 → Y lets us treat X as a subspace of Y. Every embedding is injective and continuous. Every map that is injective, continuous and either open or closed 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 Y.
Differential geometry
In differential geometry: Let M and N be smooth manifolds and <math>f:M\to N<math> be a smooth map, it is called an immersion if for any point <math>x\in M<math> the differential <math>d_f:T_x(M)\to T_{f(x)}(N)<math> is injective (here <math>T_x(M)<math> denotes tangent space of <math>M<math> at <math>x<math>). Then an embedding, or a smooth embedding, is defined to be an immersion which is an embedding in the above sense (i.e. homeomorphism onto its image). When the manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion.
In other words, an embedding is 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 <math>x\in M<math> there is a neighborhood <math>x\in U\subset M<math> such that <math>f:U\to N<math> is an embedding.)
An important case is N=Rn. The interest here is in how large n must be, in terms of the dimension m of M. The Whitney embedding theorem states that n = 2m is enough. For example the real projective plane of dimension 2 requires n = 4 for an embedding. The less restrictive condition of immersion applies to the Boy's surface—which has self-intersections.
Riemannian geometry
In Riemannian geometry: Let (M,g) and (N,h) be Riemannian manifolds. An isometric embedding is a smooth embedding f : M → N which preserves the 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
- <math>v,w\in T_x(M)<math>
we have
- <math>g(v,w)=h(df(v),df(w))<math>.
Analogously, isometric immersion is an immersion between Riemannian manifolds which preserves the Riemannian metrics.
Equivalently, an isometric embedding (immersion) is a smooth embedding (immersion) which preserves length of curves (cf. Nash embedding theorem).
Algebra
Field theory
In field theory, an embedding of a field E in a field F is a ring homomorphism σ : E → F.
The kernel of σ is an ideal of E which cannot be the whole field E, because of the condition σ(1)=1. Therefore the kernel is 0 and thus any embedding of fields is a monomorphism. Moreover, E is isomorphic to the subfield σ(E) of F. This justifies the name embedding for an arbitrary homomorphism of fields.
Domain theory
In domain theory, an embedding of partial orders is F in the function space [X → Y] such that
- For all x1, x2 in X, x1 ≤ x2 if and only if F (x1) ≤ F(x2) and
- For all y in Y, {x : F (x) ≤ y } is directed.
Based on an article from FOLDOC, used by permission.