In

^{''m''} of dimension $m$, where $m$ is a power of two, requires $n\; =\; 2m$ for an embedding. However, this does not apply to immersions; for instance, RP^{2} can be immersed in $\backslash mathbb^3$ as is explicitly shown by

Embedding of manifolds

on the Manifold Atlas {{set index article Abstract algebra Category theory General topology Differential topology Functions and mappings Maps of manifolds Model theory Order theory

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...

, an embedding (or imbedding) is one instance of some mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

contained within another instance, such as a group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

that is a subgroup
In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...

.
When some object ''X'' is said to be embedded in another object ''Y'', the embedding is given by some injective
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

and structure-preserving map . The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which ''X'' and ''Y'' are instances. In the terminology of category theory
Category theory formalizes and its concepts in terms of a called a ', whose nodes are called ''objects'', and whose labelled directed edges are called ''arrows'' (or s). A has two basic properties: the ability to the arrows , and the exi ...

, a structure-preserving map is called a morphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

.
The fact that a map is an embedding is often indicated by the use of a "hooked arrow" (); thus: $f\; :\; X\; \backslash hookrightarrow\; Y.$ (On the other hand, this notation is sometimes reserved for inclusion map
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s.)
Given ''X'' and ''Y'', several different embeddings of ''X'' in ''Y'' may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural number
File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...)
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...

s in the integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...

s, the integers in the rational number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

s, the rational numbers in the real number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s, and the real numbers in the complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s. In such cases it is common to identify the domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...

''X'' with its image
An SAR radar image acquired by the SIR-C/X-SAR radar on board the Space Shuttle Endeavour shows the Teide volcano. The city of Santa Cruz de Tenerife is visible as the purple and white area on the lower right edge of the island. Lava flows ...

''f''(''X'') contained in ''Y'', so that .
Topology and geometry

General topology

Ingeneral topology
, a useful example in point-set topology. It is connected but not path-connected.
In mathematics, general topology is the branch of topology that deals with the basic Set theory, set-theoretic definitions and constructions used in topology. It is t ...

, an embedding is a homeomorphism
and a donut (torus
In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle.
If the axis of ...

onto its image. More explicitly, an injective continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ga ...

map $f\; :\; X\; \backslash to\; Y$ between topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

s $X$ and $Y$ is a topological embedding if $f$ yields a homeomorphism between $X$ and $f(X)$ (where $f(X)$ carries the subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

inherited from $Y$). Intuitively then, the embedding $f\; :\; X\; \backslash to\; Y$ lets us treat $X$ as a of $Y$. Every embedding is injective and continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ga ...

. Every map that is injective, continuous and either open
Open or OPEN may refer to:
Music
* Open (band)
Open is a band.
Background
Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...

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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

nor a closed set
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

in $Y$.
For a given space $Y$, the existence of an embedding $X\; \backslash to\; Y$ is a topological invariantIn topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...

of $X$. This allows two spaces to be distinguished if one is able to be embedded in a space while the other is not.
Differential topology

Indifferential topology
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

:
Let $M$ and $N$ be smooth manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

s and $f:M\backslash to\; N$ be a smooth map. Then $f$ is called an immersion if its derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its Argument of a function, argument (input value). Derivatives are a fundament ...

is everywhere injective. An embedding, or a smooth embedding, is defined to be an injective immersion which is an embedding in the topological sense mentioned above (i.e. homeomorphism
and a donut (torus
In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle.
If the axis of ...

onto its image).
In other words, the domain of an embedding is diffeomorphic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

to its image, and in particular the image of an embedding must be a submanifold
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

. An immersion is precisely a local embedding, i.e. for any point $x\backslash in\; M$ there is a neighborhood $x\backslash in\; U\backslash subset\; M$ such that $f:U\backslash to\; N$ 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 $N\; =\; \backslash mathbb^n$. The interest here is in how large $n$ must be for an embedding, in terms of the dimension $m$ of $M$. The Whitney embedding theorem
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

states that $n\; =\; 2m$ is enough, and is the best possible linear bound. For example, the real projective spaceIn mathematics, real projective space, or RP''n'' or \mathbb_n(\mathbb), is the topological space of lines passing through the origin 0 in R''n''+1. It is a compact space, compact, smooth manifold of dimension ''n'', and is a special case Gr(1, R''n' ...

RPBoy's surface
In geometry, Boy's surface is an immersion (mathematics), immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901. He discovered it on assignment from David Hilbert to prove that the projective plane ''could not'' ...

—which has self-intersections. The Roman surface
The Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion (mathematics), immersion of the projective ...

fails to be an immersion as it contains cross-cap
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s.
An embedding is proper if it behaves well with respect to boundaries: one requires the map $f:\; X\; \backslash rightarrow\; Y$ to be such that
*$f(\backslash partial\; X)\; =\; f(X)\; \backslash cap\; \backslash partial\; Y$, and
*$f(X)$ is transverse
Transverse may refer to:
*Transverse engine, an engine in which the crankshaft is oriented side-to-side relative to the wheels of the vehicle
*Transverse flute, a flute that is held horizontally
*Euler force, Transverse force (or ''Euler force''), ...

to $\backslash partial\; Y$ in any point of $f(\backslash partial\; X)$.
The first condition is equivalent to having $f(\backslash partial\; X)\; \backslash subseteq\; \backslash partial\; Y$ and $f(X\; \backslash setminus\; \backslash partial\; X)\; \backslash subseteq\; Y\; \backslash setminus\; \backslash partial\; Y$. The second condition, roughly speaking, says that ''f''(''X'') is not tangent to the boundary of ''Y''.
Riemannian and pseudo-Riemannian geometry

InRiemannian geometry#REDIRECT Riemannian geometry
Riemannian geometry is the branch of differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and m ...

and pseudo-Riemannian geometry:
Let (''M'', ''g'') and (''N'', ''h'') be Riemannian manifold
In differential geometry
Differential geometry is a mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a c ...

s or more generally pseudo-Riemannian manifold
In differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differenti ...

s.
An isometric embedding is a smooth embedding ''f'' : ''M'' → ''N'' which preserves the (pseudo-)metric
METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...

in the sense that ''g'' is equal to the pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a Pushforward (disambiguation), pushforward.
Precomposition
Precomposition with a Function (mathematics), function probabl ...

of ''h'' by ''f'', i.e. ''g'' = ''f''*''h''. Explicitly, for any two tangent vectors $v,w\backslash in\; T\_x(M)$ we have
:$g(v,w)=h(df(v),df(w)).$
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 curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

s (cf. Nash embedding theorem
The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space
Euclidean space is the fundamental space of classical geometry. Ori ...

).Nash J., ''The embedding problem for Riemannian manifolds,'' Ann. of Math. (2), 63 (1956), 20–63.
Algebra

In general, for an algebraic category ''C'', an embedding between two ''C''-algebraic structures ''X'' and ''Y'' is a ''C''-morphism that is injective.Field theory

In field theory, an embedding of afield
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

''E'' in a field ''F'' is a ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function (mathematics), function between two ring (algebra), rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function s ...

.
The kernel
Kernel may refer to:
Computing
* Kernel (operating system)
In an operating system with a Abstraction layer, layered architecture, the kernel is the lowest level, has complete control of the hardware and is always in memory. In some systems it ...

of ''σ'' is an ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...

of ''E'' which cannot be the whole field ''E'', because of the condition . 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
220px
In the context of abstract algebra or universal algebra, a monomorphism is an Injective function, injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of catego ...

. Hence, ''E'' is isomorphic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

to the subfield ''σ''(''E'') of ''F''. This justifies the name ''embedding'' for an arbitrary homomorphism of fields.
Universal algebra and model theory

If σ is asignature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a s ...

and $A,B$ are σ-structures
A structure is an arrangement and organization of interrelated elements in a material object or system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A sy ...

(also called σ-algebras in universal algebraUniversal algebra (sometimes called general algebra) is the field of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geomet ...

or models in model theory
In mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...

), then a map $h:A\; \backslash to\; B$ is a σ-embedding iff all of the following hold:
* $h$ is injective,
* for every $n$-ary function symbol $f\; \backslash in\backslash sigma$ and $a\_1,\backslash ldots,a\_n\; \backslash in\; A^n,$ we have $h(f^A(a\_1,\backslash ldots,a\_n))=f^B(h(a\_1),\backslash ldots,h(a\_n))$,
* for every $n$-ary relation symbol $R\; \backslash in\backslash sigma$ and $a\_1,\backslash ldots,a\_n\; \backslash in\; A^n,$ we have $A\; \backslash models\; R(a\_1,\backslash ldots,a\_n)$ iff $B\; \backslash models\; R(h(a\_1),\backslash ldots,h(a\_n)).$
Here $A\backslash models\; R\; (a\_1,\backslash ldots,a\_n)$ is a model theoretical notation equivalent to $(a\_1,\backslash ldots,a\_n)\backslash in\; R^A$. In model theory there is also a stronger notion of elementary embeddingIn model theory
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis ...

.
Order theory and domain theory

Inorder theory
Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intro ...

, an embedding of partially ordered set
upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not.
In mathem ...

s is a function ''F'' between partially ordered sets ''X'' and ''Y'' such that
:$\backslash forall\; x\_1,x\_2\backslash in\; X:\; x\_1\backslash leq\; x\_2\; \backslash iff\; F(x\_1)\backslash leq\; F(x\_2).$
Injectivity of ''F'' follows quickly from this definition. In domain theory
Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer s ...

, an additional requirement is that
:$\backslash forall\; y\backslash in\; Y:\backslash $ is directed
Director may refer to:
Literature
* Director (magazine), ''Director'' (magazine), a British magazine
* The Director (novel), ''The Director'' (novel), a 1971 novel by Henry Denker
* The Director (play), ''The Director'' (play), a 2000 play by Nanc ...

.
Metric spaces

A mapping $\backslash phi:\; X\; \backslash to\; Y$ ofmetric spaces
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
Mathematics
* Metric (mathematics), an abstraction of the notion of ''distance'' in a metric space
* Metric tensor, in differential geometr ...

is called an ''embedding''
(with distortion
Distortion is the alteration of the original shape (or other characteristic) of something. In and it means the alteration of the of an information-bearing , such as an representing sound or a representing images, in an electronic device or ...

$C>0$) if
:$L\; d\_X(x,\; y)\; \backslash leq\; d\_Y(\backslash phi(x),\; \backslash phi(y))\; \backslash leq\; CLd\_X(x,y)$
for every $x,y\backslash in\; X$ and some constant $L>0$.
Normed spaces

An important special case is that ofnormed spaces
The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk
Dunkirk (, ; french: Dunkerque ; vls, label=French Flemish, Duunkerke; nl, Duinkerke(n) ) is a Communes of France, ...

; 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

$(X,\; \backslash ,\; \backslash cdot\; \backslash ,\; )$ is, ''what is the maximal dimension $k$ such that the Hilbert space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

$\backslash ell\_2^k$ can be linearly embedded into $X$ with constant distortion?''
The answer is given by Dvoretzky's theoremIn mathematics, Dvoretzky's theorem is an important structural theorem about normed vector spaces proved by Aryeh Dvoretzky in the early 1960s, answering a question of Alexander Grothendieck. In essence, it says that every sufficiently high-dimension ...

.
Category theory

Incategory theory
Category theory formalizes and its concepts in terms of a called a ', whose nodes are called ''objects'', and whose labelled directed edges are called ''arrows'' (or s). A has two basic properties: the ability to the arrows , and the exi ...

, there is no satisfactory and generally accepted definition of embeddings that is applicable in all categories. One would expect that all isomorphisms and all compositions of embeddings are embeddings, and that all embeddings are monomorphisms. Other typical requirements are: any extremal monomorphism is an embedding and embeddings are stable under pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a Pushforward (disambiguation), pushforward.
Precomposition
Precomposition with a Function (mathematics), function probabl ...

s.
Ideally the class of all embedded subobject In category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), sp ...

s of a given object, up to isomorphism, should also be small
Small may refer to:
Science and technology
* SMALL
Small may refer to:
Science and technology
* SMALL
Small may refer to:
Science and technology
* SMALL
Small may refer to:
Science and technology
* SMALL, an ALGOL-like programming language ...

, and thus an ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...

. In this case, the category is said to be well powered with respect to the class of embeddings. This allows defining new local structures in the category (such as a closure operatorIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

).
In a concrete category
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

, an embedding is a morphism ''ƒ'': ''A'' → ''B'' which is an injective function from the underlying set of ''A'' to the underlying set of ''B'' and is also an initial morphism in the following sense:
If ''g'' is a function from the underlying set of an object ''C'' to the underlying set of ''A'', and if its composition with ''ƒ'' is a morphism ''ƒg'': ''C'' → ''B'', then ''g'' itself is a morphism.
A factorization system for a category also gives rise to a notion of embedding. If (''E'', ''M'') is a factorization system, then the morphisms in ''M'' may be regarded as the embeddings, especially when the category is well powered with respect to ''M''. Concrete theories often have a factorization system in which ''M'' consists of the embeddings in the previous sense. This is the case of the majority of the examples given in this article.
As usual in category theory, there is a dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
** . . . see more cases in :Duality theories
* Dual ...

concept, known as quotient. All the preceding properties can be dualized.
An embedding can also refer to an embedding functor.
See also

*Closed immersion:''For the same-name concept in differential geometry, see immersion (mathematics).''
In algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algeb ...

*Cover
Cover or covers may refer to:
Packaging, science and technology
* A covering, usually - but not necessarily - one that completely closes the object
** Cover (philately), generic term for envelope or package
** Housing (engineering), an exterior ...

*Dimension reduction
Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally ...

* Immersion
*Johnson–Lindenstrauss lemmaIn mathematics, the Johnson–Lindenstrauss lemma is a result named after William B. Johnson (mathematician), William B. Johnson and Joram Lindenstrauss concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclide ...

*Submanifold
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

* Subspace
* Universal space
Notes

References

* * * * * * * * * * * * . * * .External links

*Embedding of manifolds

on the Manifold Atlas {{set index article Abstract algebra Category theory General topology Differential topology Functions and mappings Maps of manifolds Model theory Order theory