Embedding

In general topology, an embedding is a homeomorphism onto its image.[3] More explicitly, an injective continuous map ${\displaystyle f:X\to Y}$ between topological spaces ${\displaystyle X}$ and ${\displaystyle Y}$ is a topological embedding if ${\displaystyle f}$ yields a homeomorphism between ${\displaystyle X}$ and ${\displaystyle f(X)}$ (where ${\displaystyle f(X)}$ carries the subspace topology inherited from ${\displaystyle Y}$). Intuitively then, the embedding ${\displaystyle f:X\to Y}$ lets us treat ${\displaystyle X}$ as a subspace of ${\displaystyle Y}$. 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 ${\displaystyle f(X)}$ is neither an open set nor a closed set in ${\displaystyle Y}$.

For a given space ${\displaystyle Y}$, the existence of an embedding ${\displaystyle X\to Y}$ is a topological invariant of ${\displaystyle X}$. This allows two spaces to be distinguished if one is able to be embedded in a space while the other is not.

If the domain of a function ${\displaystyle f:X\to Y}$ is a topological space then the function is said to be locally injective at a point if there exists some neighborhood ${\displaystyle U}$ of this point such that the restriction ${\displaystyle f{\big \vert }_{U}:U\to Y}$ is injective. It is called locally injective if it is locally injective around every point of its domain. Similarly, a local (topological, resp. smooth) embedding is a function for which every point in its domain has some neighborhood to which its restriction is a (topological, resp. smooth) embedding.

Every injective function is locally injective but not conversely. Local diffeomorphisms, local homeomorphisms, and smooth immersions are all locally injective functions that are not necessarily injective. The inverse function theorem gives a sufficient condition for a continuously differentiable function to be (among other things) locally injective. Every fiber of a locally injective function ${\displaystyle f:X\to Y}$ is necessarily a discrete subspace of its domain ${\displaystyle X.}$

In differential topology: Let ${\displaystyle M}$ and ${\displaystyle N}$ be smooth manifolds and ${\displaystyle f:M\to N}$ be a smooth map. Then ${\displaystyle f}$ is called an immersion if its derivative is everywhere injective. An embedding, or a smooth embedding, is defined to be an immersion which is an embedding in the topological sense mentioned above (i.e. homeomorphism onto its image).[4]

In other words, the domain of an embedding is diffeomorphic to its image, and in particular the image of an embedding must be a submanifold. An immersion is precisely a local embedding, i.e. for any point ${\displaystyle x\in M}$ there is a neighborhood ${\displaystyle x\in U\subset M}$ such that ${\displaystyle f:U\to N}$ is an embedding.

When the domain manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion.

An important case is ${\displaystyle N=\mathbb {R} ^{n}}$. The interest here is in how large ${\displaystyle n}$ must be for an embedding, in terms of the dimension ${\displaystyle m}$ of ${\displaystyle M}$. The Whitney embedding theorem[5] states that ${\displaystyle n=2m}$ is enough, and is the best possible linear bound. For example, the real projective space ${\displaystyle RP^{m}}$ of dimension ${\displaystyle m}$, where ${\displaystyle m}$ is a power of two, requires ${\displaystyle n=2m}$ for an embedding. However, this does not apply to immersions; for instance, ${\displaystyle RP^{2}}$ can be immersed in ${\displaystyle \mathbb {R} ^{3}}$ as is explicitly shown by Boy's surface—which has self-intersections. The Roman surface fails to be an immersion as it contains cross-caps.

An embedding is proper if it behaves well with respect to boundaries: one requires the map ${\displaystyle f:X\rightarrow Y}$ to be such that

• ${\displaystyle f(\partial X)=f(X)\cap \partial Y}$, and
• ${\displaystyle f(X)}$ is transverse to ${\displaystyle \partial Y}$ in any point of ${\displaystyle f(\partial X)}$.

The first condition is equivalent to having ${\displaystyle f(\partial X)\subseteq \partial Y}$ and ${\displaystyle f(X\setminus \partial X)\subseteq Y\setminus \partial Y}$. The second condition, roughly speaking, says that ${\displaystyle f(X)}$ is not tangent to the boundary of ${\displaystyle Y}$.

In Riemannian geometry and pseudo-Riemannian geometry: Let ${\displaystyle (M,g)}$ and ${\displaystyle (N,h)}$ be Riemannian manifolds or more generally pseudo-Riemannian manifolds. An isometric embedding is a smooth embedding ${\displaystyle f:M\rightarrow N}$ which preserves the (pseudo-)metric in the sense that ${\displaystyle g}$ is equal to the pullback of ${\displaystyle h}$ by ${\displaystyle f}$, i.e. ${\displaystyle g=f^{*}h}$. Explicitly, for any two tangent vectors ${\displaystyle v,w\in T_{x}(M)}$ we have

${\displaystyle g(v,w)=h(df(v),df(w)).}$

Analogously, isometric immersion is an immersion between (pseudo)-Riemannian manifolds which preserves the (pseudo)-Riemannian metrics.

Equivalently, in Riemannian geometry, an isometric embedding (immersion) is a smooth embedding (immersion) which preserves length of curves (cf. Nash embedding theorem).[6]

In field theory, an embedding of a field ${\displaystyle E}$ in a field ${\displaystyle F}$ is a ring homomorphism ${\displaystyle \sigma :E\rightarrow F}$.

The kernel of ${\displaystyle \sigma }$ is an ideal of ${\displaystyle E}$ which cannot be the whole field ${\displaystyle E}$, because of the condition ${\displaystyle 1=\sigma (1)=1}$. 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 ${\displaystyle 0}$, so any embedding of fields is a monomorphism. Hence, ${\displaystyle E}$ is isomorphic to the subfield ${\displaystyle \sigma (E)}$ of ${\displaystyle F}$. This justifies the name embedding for an arbitrary homomorphism of fields.

If ${\displaystyle \sigma }$ is a signature and ${\displaystyle A,B}$ are ${\displaystyle \sigma }$-structures (also called ${\displaystyle \sigma }$-algebras in universal algebra or models in model theory), then a map ${\displaystyle h:A\to B}$ is a ${\displaystyle \sigma }$-embedding exactly if all of the following hold:

• ${\displaystyle h}$ is injective,
• for every ${\displaystyle n}$-ary function symbol ${\displaystyle f\in \sigma }$ and ${\displaystyle a_{1},\ldots ,a_{n}\in A^{n},}$ we have ${\displaystyle h(f^{A}(a_{1},\ldots ,a_{n}))=f^{B}(h(a_{1}),\ldots ,h(a_{n}))}$,
• for every ${\displaystyle n}$-ary relation symbol ${\displaystyle R\in \sigma }$ and ${\displaystyle a_{1},\ldots ,a_{n}\in A^{n},}$ we have ${\displaystyle A\models R(a_{1},\ldots ,a_{n})}$ iff ${\displaystyle B\models R(h(a_{1}),\ldots ,h(a_{n})).}$

Here ${\displaystyle A\models R(a_{1},\ldots ,a_{n})}$ is a model theoretical notation equivalent to ${\displaystyle (a_{1},\ldots ,a_{n})\in R^{A}}$. In model theory there is also a stronger notion of elementary embedding.

An important special case is that of normed spaces; in this case it is natural to consider linear embeddings.

One of the basic questions that can be asked about a finite-dimensional normed space ${\displaystyle (X,\|\cdot \|)}$ is, what is the maximal dimension ${\displaystyle k}$ such that the Hilbert space ${\displaystyle \ell _{2}^{k}}$ can be linearly embedded into ${\displaystyle X}$ with constant distortion?

The answer is given by Dvoretzky's theorem.