Embedding
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- For other uses of this term, see embedded (disambiguation).
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.
For a given space X, the existence of an embedding X → Y is a topological invariant of X. This allows two spaces to be distinguished if one is able to be embedded into a space which the other is not.
An embedding is proper if it behaves well w.r.t. 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 transversal to [\partial Y] in any point of [f(\partial X)].
Differential geometry
In differential geometry: Let M and N be smooth manifolds and [f:M\to N] be a smooth map, it is called an immersion if for any point [x\in M] the differential [d_xf:T_x(M)\to T_(N)] is injective (here [T_x(M)] denotes tangent space of [M] at [x]). 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 [x\in M] there is a neighborhood [x\in U\subset M] such that [f:U\to N] 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. The Roman surface fails to be an immersion as it contains cross-caps.
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
- [v,w\in T_x(M)]
- [g(v,w)=h(df(v),df(w))].
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. 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. 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
- [\forall x_1,x_2\in X: x_1\leq x_2\Leftrightarrow F(x_1)\leq F(x_2)] and
- [ \forall y\in Y:\] is directed.
See also
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