Embedding

When some object ${\displaystyle X}$ is said to be embedded in another object ${\displaystyle Y}$, the embedding is given by some injective and structure-preserving map ${\displaystyle f:X\to Y}$. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which ${\displaystyle X}$ and ${\displaystyle Y}$ are instances. In the terminology of category theory, a structure-preserving map is called a morphism.

The fact that a map ${\displaystyle f:X\to Y}$ is an embedding is often indicated by the use of a "hooked arrow" (U+21AA ↪ RIGHTWARDS ARROW WITH HOOK); thus: ${\displaystyle f:X\hookrightarrow Y.}$ (On the other hand, this notation is sometimes reserved for inclusion maps.)

Given ${\displaystyle X}$ and ${\displaystyle Y}$, several different embeddings of ${\displaystyle X}$ in ${\displaystyle Y}$ may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers in the rational numbers, the rational numbers in the real numbers, and the real numbers in the complex numbers. In such cases it is common to identify the domain ${\displaystyle X}$ with its image ${\displaystyle f(X)}$ contained in ${\displaystyle Y}$, so that ${\displaystyle X\subseteq Y}$.

Topology and geometry

General topology

In general topology, an embedding is a homeomorphism onto its image. 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 that 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.

Related definitions

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 ${\displaystyle f{|}_U:U\to Y}$ 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 ${\displaystyle X.}$

Differential topology

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 that is an embedding in the topological sense mentioned above (i.e. homeomorphism onto its image).

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 states that ${\displaystyle n=2m}$ is enough, and is the best possible linear bound. For example, the real projective space ${\displaystyle \mathbb{R}{\mathrm{P}}^m}$ 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, ${\displaystyle \mathbb{R}{\mathrm{P}}^2}$ 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 ${\displaystyle f:X\to Y}$ to be such that

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

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

Riemannian and pseudo-Riemannian geometry

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 ${\displaystyle f:M\to N}$ that preserves the (pseudo-)metric in the sense that ${\displaystyle g}$ is equal to the pullback of ${\displaystyle h}$ by ${\displaystyle f}$, i.e. ${\displaystyle g=f^{\ast }h}$. 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 that preserves the (pseudo)-Riemannian metrics.

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