# Embedding

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 : XY 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 : XY 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 [itex]f:M\to N[itex] be a smooth map, it is called an immersion if for any point [itex]x\in M[itex] the differential [itex]d_f:T_x(M)\to T_{f(x)}(N)[itex] is injective (here [itex]T_x(M)[itex] denotes tangent space of [itex]M[itex] at [itex]x[itex]). 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 [itex]x\in M[itex] there is a neighborhood [itex]x\in U\subset M[itex] such that [itex]f:U\to N[itex] 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 : MN 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

[itex]v,w\in T_x(M)[itex]

we have

[itex]g(v,w)=h(df(v),df(w))[itex].

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 : EF.

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

1. For all x1, x2 in X, x1 ≤ x2 if and only if F (x1) ≤ F(x2) and
2. For all y in Y, {x : F (x) ≤ y } is directed.

Based on an article from FOLDOC, used by permission.

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