Topological Embedding

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 .

ABSTRACTLY OR CATEGORICALLY

An abstract embedding between two

X,Y\,

Objects in a given Category

\mathfrak{C}

, is a

\mathfrak{C}

- Morphism

f\colon X	o Y

which is Injective .

TOPOLOGY AND GEOMETRY

General topology

In 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 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

x\in M

there is a neighborhood

x\in U\subset 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''=Rⁿ. 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 R³, 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 Manifold s.

''h''. Explicitly, for any two tangent vectors

v,w\in 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 Curve s (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 Order s is

F

in the Function Space →Y such that

#

orall x_1,x_2\in X: x_1\leq x_2\Leftrightarrow F(x_1)\leq F(x_2)

and
#

orall y\in Y:\{x: F(x)\leq y\}

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.

	CATEGORIES ABOUT EMBEDDING

	category theory

	general topology

	differential geometry

	differential topology

	order theory

	abstract algebra

	model theory

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Topological Embedding