Embedding

In various areas of mathematics is meant by embedding an image, which allows an object to be construed as part of another.

Often, so it is meant only an injective mapping or a monomorphism. For example, we speak of the canonical embedding of the real numbers to complex numbers.

Moreover, there are in some areas more specific embedding terms.

Topology

In the topology is called a mapping between two topological spaces and embedding as in, if a homeomorphism of onto the subspace of its image is ( in the subspace topology).

There are the following statements are equivalent:

• The mapping is an embedding.
• Is injective, continuous and as picture after open, ie for every open set of the image is open again in.
• Is injective and continuous, and for all topological spaces and all continuous maps that factorize over (ie there is a picture with ), the induced map is continuous.
• Is an extreme monomorphism, i.e., is injective for each factorization into an epimorphism (ie a surjective continuous map ) and a continuous map, is not only a Bimorphismus (ie bijective ) as for any injective, but even a homeomorphism.
• Is a regular monomorphism.

In general, an embedding is not open, ie for open must not be open in, as the example of the usual embedding shows. Is an embedding if and only open when the image is open.

Differential topology

Under a smooth embedding is defined as a topological embedding a differentiable manifold in a differentiable manifold, which is also still an immersion.

Differential Geometry

Under an isometric embedding of a Riemannian manifold into a Riemannian manifold is defined as a smooth embedding in, such that for all tangent vectors in the equation.

An isometric embedding obtains the lengths of curves, but it does not necessarily get the distances between points. As an example, consider the one with the Euclidean metric and the unit sphere with the induced metric. By definition of the induced metric, the inclusion is an isometric embedding. But it is not distances - preserving: for example, the distance between the north and south pole (ie the length of a shortest connecting curve ) on the same, while their distance in the same.