EMBEDDING

:''For other uses of this term, see embedded (disambiguation).''
In mathematics, an 'embedding' (or 'imbedding') is one instance of some mathematical object contained within another instance, such as a group that is a subgroup.

Contents

Abstractly or categorically

Topology and Geometry

General topology

Differential topology

Riemannian geometry

Algebra

Field theory

Domain theory

Metric spaces

Normed spaces

Model theory

See also

Abstractly or categorically

An 'abstract embedding' between two $X,Y,$ objects in a given category $mathfrak\{C\}$, is a $mathfrak\{C\}$-morphism $fcolon\; X\; o\; Y$ which is injective.

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

Differential topology

In differential topology:
Let ''M'' and ''N'' be smooth manifolds and $f:M\; o\; N$ be a smooth map, it is called an immersion if the derivative of ''f'' is everywhere injective. 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).
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 $xin\; M$ there is a neighborhood $xin\; Usubset\; M$ such that $f:U\; o\; N$ is an embedding.)
When the manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion.
An important case is ''N''='R'ⁿ. 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'' = 2''m'' is enough. For example the real projective plane of dimension 2 requires ''n'' = 4 for an embedding. An immersion of this surface is, however, possible in 'R'³, and one example is 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
:$v,win\; T\_x(M)$
we have
:$g(v,w)=h(df(v),df(w)).,$
Analogously, 'isometric immersion' is an immersion between Riemannian manifolds which preserves the Riemannian metrics.
Equivalently, an isometric embedding (immersion) is a smooth embedding (immersion) which preserves length of curves (cf. Nash embedding theorem).

Algebra

In general, for a category ''C'', an embedding between two ''C''-algebraic structures ''X'' and ''Y'' is a ''C''-morphism ''e:X→Y'' which is injective.

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
# and
# is directed.
''Based on an article from FOLDOC, .''

Metric spaces

A mapping $phi:\; X\; o\; Y$ of metric spaces is called an ''embedding''
(with distortion $C>0$) if
:$L\; d\_X(x,\; y)\; leq\; d\_Y(phi(x),\; phi(y))\; leq\; CLd\_X(x,y)$
for some constant $L>0$.

Normed spaces

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 $(X,\; |\; cdot\; |)$ is, ''what is the maximal dimension $k$ such that the Hilbert space $ell\_2^k$ can be linearly embedded into $X$ with constant distortion?''
The answer is given by Dvoretzky's theorem.

Model theory

If $L$ is a first order language and $A,B$ are $L$-structures, then a map $sigma:A\; o\; B$ is an $L$-embedding iff all the following holds:

★ $sigma$ is injective,

★ for every n-ary function symbol $f\; in\; L$ and $a\_1,ldots,a\_n\; in\; A^n$, we have $sigma(f^A(a\_1,ldots,a\_n))=f^B(sigma(a\_1),ldots,sigma(a\_n))$,

★ for every n-ary relation symbol $R\; in\; L$ and $a\_1,ldots,a\_n\; in\; A^n,$ we have $A\; models\; R(a\_1,ldots,a\_n)$ iff $B\; models\; R(sigma(a\_1),ldots,sigma(a\_n)),$

★ for every constant symbol $c\; in\; L$, $sigma(c^A)=c^B$.

See also

★ Inclusion map