Embedding (mathematics)

In various areas of mathematics , embedding is understood to be a mapping that enables one object to be understood as part of another.

Often only an injective mapping or a monomorphism is meant. For example, one speaks of the canonical embedding of the real numbers in the complex numbers .

In addition, there are more specific embedding terms in some areas.

topology

In topology one describes a mapping between two topological spaces and as embedding of in , if a homeomorphism is from on the subspace of its image (in the subspace topology ). ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle f(X)}$

The following statements are equivalent:

• the image is an embedding. ${\displaystyle f\colon X\rightarrow Y}$
• ${\displaystyle f}$is injective, continuous and as a mapping to open, d. i.e., for each open set of , the image is open in again . ${\displaystyle f(X)}$ ${\displaystyle O}$ ${\displaystyle X}$ ${\displaystyle f(O)}$ ${\displaystyle f(X)}$
• ${\displaystyle f}$is injective and continuous, and for all topological spaces and all continuous maps that factorize over (i.e. there is a map with ) the induced map is continuous. ${\displaystyle T}$ ${\displaystyle t\colon T\rightarrow Y}$ ${\displaystyle X}$ ${\displaystyle t_{0}\colon T\rightarrow X}$ ${\displaystyle t=f\circ t_{0}}$ ${\displaystyle t_{0}\ }$
• ${\displaystyle f}$is an extreme monomorphism , i. H. is injective for each factorization in a epimorphism (d. e. a surjective continuous mapping) and a continuous map , is not only a Bimorphismus (d. h. bijective) as for any injective , but even a homeomorphism. ${\displaystyle f}$ ${\displaystyle e}$ ${\displaystyle g}$ ${\displaystyle f=g\circ e}$ ${\displaystyle e}$ ${\displaystyle f}$
• ${\displaystyle f}$is a regular monomorphism .

In general, embedding is not open; That is, for open does not have to be open in , as the example of the usual embedding shows. An embedding is open exactly when the image in is open. ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle U\subset X}$ ${\displaystyle f(U)}$ ${\displaystyle Y}$ ${\displaystyle f\colon \mathbb {R} \to \mathbb {C} }$ ${\displaystyle f}$ ${\displaystyle f(X)}$ ${\displaystyle Y}$

Differential topology

A smooth embedding is understood to mean a topological embedding of a differentiable manifold in a differentiable manifold , which is also an immersion . ${\displaystyle X}$ ${\displaystyle Y}$

Differential geometry

Under an isometric embedding a Riemannian manifold into a Riemannian manifold is meant a smooth embedding of in , such that for all tangent vectors in the equation is valid. ${\displaystyle (X,g_{1})}$ ${\displaystyle (Y,g_{2})}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle v,w}$ ${\displaystyle T_{x}X}$ ${\displaystyle g_{2}(Df(v),Df(w))=g_{1}(v,w)}$

An isometric embedding preserves the lengths of curves , but does not necessarily have to preserve the distances between points. As an example, consider the one with the Euclidean metric and the unit sphere with the induced metric. According to the definition of the induced metric, the inclusion is an isometric embedding. However, it is not distance-maintaining: for example, the distance between the north and south poles (i.e. the length of a shortest connecting curve) is the same on the while their distance im is the same . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle S^{n-1}\subset \mathbb {R} ^{n}}$ ${\displaystyle S^{n-1}\rightarrow \mathbb {R} ^{n}}$ ${\displaystyle S^{n-1}}$ ${\displaystyle \pi }$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle 2}$

Body theory

In body theory , every nontrivial ring homomorphism is already a body embedding , i.e. a monomorphism . ${\displaystyle E\to F}$

A number field can have different embeddings . An embedding is called a real embedding if your image is in , and complex embedding otherwise. For example, one has real and two complex embeddings. (The complex embeddings map to the other zeros of .) For every complex embedding, the complex-conjugate delivers a different complex embedding, which is why the number of complex embeddings is always even. The following applies , where denotes the number of real and the number of complex embeddings. ${\displaystyle K\subset \mathbb {C} }$ ${\displaystyle K\subset \mathbb {C} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}})}$ ${\displaystyle {\sqrt[{3}]{2}}}$ ${\displaystyle x^{3}-2}$ ${\displaystyle \left[K:\mathbb {Q} \right]=r_{1}+2r_{2}}$ ${\displaystyle r_{1}}$ ${\displaystyle 2r_{2}}$

See also

• Subspace
• Actual embedding
• Mitchell embedding theorem
• Nash embedding theorem
• Sobolev's embedding theorem
• Whitney embedding theorem

Individual evidence

1. ↑ extremal monomorphism , entry in the nLab . (English)

literature

• Boto von Querenburg : Set theoretical topology (= Springer textbook ). 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9 .

Web links

Wiktionary: embedding  - explanations of meanings, word origins, synonyms, translations