topological embedding

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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an embedding (or imbedding) is one instance of some
mathematical structure In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additio ...
contained within another instance, such as a group that is a
subgroup In group theory, a branch of mathematics, given a 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, ''H'' is a subgrou ...
. When some object $X$ is said to be embedded in another object $Y$, the embedding is given by some injective and structure-preserving map $f:X\rightarrow Y$. 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 is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...
, a structure-preserving map is called a
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphism ...
. The fact that a map $f:X\rightarrow Y$ is an embedding is often indicated by the use of a "hooked arrow" (); thus: $f : X \hookrightarrow Y.$ (On the other hand, this notation is sometimes reserved for inclusion maps.) 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 In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
s in the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, the integers in the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
s, the rational numbers in the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s, and the real numbers in the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s. In such cases it is common to identify the domain $X$ with its
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
$f\left(X\right)$ contained in $Y$, so that $f\left(X\right)\subseteq Y$.

# Topology and geometry

## General topology

In
general topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geome ...
, an embedding is a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomo ...
onto its image. More explicitly, an injective continuous map $f : X \to Y$ between
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
s $X$ and $Y$ is a topological embedding if $f$ yields a homeomorphism between $X$ and $f\left(X\right)$ (where $f\left(X\right)$ carries the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
inherited from $Y$). Intuitively then, the embedding $f : X \to Y$ 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 $f\left(X\right)$ is neither an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are s ...
nor a
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
in $Y$. For a given space $Y$, the existence of an embedding $X \to Y$ is a
topological invariant In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spa ...
of $X$. 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 $f : X \to Y$ is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
then the function is said to be ' if there exists some
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural ...
$U$ of this point such that the restriction $f\big\vert_U : U \to Y$ is injective. It is called ' if it is locally injective around every point of its domain. Similarly, a ' 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 In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its ''derivative is continuous and non-zero at t ...
gives a sufficient condition for a continuously differentiable function to be (among other things) locally injective. Every fiber of a locally injective function $f : X \to Y$ is necessarily a discrete subspace of its domain $X.$

## Differential topology

In
differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
: Let $M$ and $N$ be smooth
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
s and $f:M\to N$ 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 (input value). Derivatives are a fundamental tool of calculus. ...
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 In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomo ...
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 In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
. An immersion is precisely a local embedding, i.e. for any point $x\in M$ there is a neighborhood $x\in U\subset M$ such that $f:U\to N$ 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 $N = \mathbb^n$. 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, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney: *The strong Whitney embedding theorem states that any smooth real - dimensional manifold (required also to be Hausd ...
states that $n = 2m$ is enough, and is the best possible linear bound. For example, the
real projective space In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space. Basic properties Construction ...
$RP^m$ 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 $\mathbb^3$ 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 $f: X \rightarrow Y$ to be such that *$f\left(\partial X\right) = f\left(X\right) \cap \partial Y$, and *$f\left(X\right)$ is transverse to $\partial Y$ in any point of $f\left(\partial X\right)$. The first condition is equivalent to having $f\left(\partial X\right) \subseteq \partial Y$ and $f\left(X \setminus \partial X\right) \subseteq Y \setminus \partial Y$. The second condition, roughly speaking, says that $f\left(X\right)$ is not tangent to the boundary of $Y$.

## Riemannian and pseudo-Riemannian geometry

In Riemannian geometry and pseudo-Riemannian geometry: Let $\left(M,g\right)$ and $\left(N,h\right)$ be
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ' ...
s or more generally
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
s. An isometric embedding is a smooth embedding $f:M\rightarrow N$ which preserves the (pseudo-) 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 $v,w\in T_x\left(M\right)$ we have :$g\left(v,w\right)=h\left(df\left(v\right),df\left(w\right)\right).$ 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, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s (cf. Nash embedding theorem).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 a field $E$ in a field $F$ is a ring homomorphism . The
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
of $\sigma$ 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 considere ...
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 In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphi ...
. Hence, $E$ is isomorphic to the subfield $\sigma\left(E\right)$ of $F$. This justifies the name ''embedding'' for an arbitrary homomorphism of fields.

## Universal algebra and model theory

If $\sigma$ is a signature and $A,B$ are $\sigma$- structures (also called $\sigma$-algebras in universal algebra or models in model theory), then a map $h:A \to B$ is a $\sigma$-embedding iff all of the following hold: * $h$ is injective, * for every $n$-ary function symbol $f \in\sigma$ and $a_1,\ldots,a_n \in A^n,$ we have $h\left(f^A\left(a_1,\ldots,a_n\right)\right)=f^B\left(h\left(a_1\right),\ldots,h\left(a_n\right)\right)$, * for every $n$-ary relation symbol $R \in\sigma$ and $a_1,\ldots,a_n \in A^n,$ we have $A \models R\left(a_1,\ldots,a_n\right)$ iff $B \models R\left(h\left(a_1\right),\ldots,h\left(a_n\right)\right).$ Here $A\models R \left(a_1,\ldots,a_n\right)$ is a model theoretical notation equivalent to $\left(a_1,\ldots,a_n\right)\in R^A$. In model theory there is also a stronger notion of elementary embedding.

# Order theory and domain theory

In order theory, an embedding of
partially 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. A poset consists of a set together with a binary ...
s is a function $F$ between partially ordered sets $X$ and $Y$ such that :$\forall x_1,x_2\in X: x_1\leq x_2 \iff F\left(x_1\right)\leq F\left(x_2\right).$ Injectivity of $F$ follows quickly from this definition. In domain theory, an additional requirement is that :$\forall y\in Y:\$ is
directed Director may refer to: Literature * ''Director'' (magazine), a British magazine * ''The Director'' (novel), a 1971 novel by Henry Denker * ''The Director'' (play), a 2000 play by Nancy Hasty Music * Director (band), an Irish rock band * ''D ...
.

# Metric spaces

A mapping $\phi: X \to Y$ of
metric spaces In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
is called an ''embedding'' (with
distortion In signal processing, distortion is the alteration of the original shape (or other characteristic) of a signal. In communications and electronics it means the alteration of the waveform of an information-bearing signal, such as an audio sig ...
$C>0$) if :$L d_X\left(x, y\right) \leq d_Y\left(\phi\left(x\right), \phi\left(y\right)\right) \leq CLd_X\left(x,y\right)$ for every $x,y\in X$ and 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 In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
$\left(X, \, \cdot \, \right)$ is, ''what is the maximal dimension $k$ such that the
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
$\ell_2^k$ can be linearly embedded into $X$ with constant distortion?'' The answer is given by Dvoretzky's theorem.

# Category theory

In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...
, 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 pullbacks. Ideally the class of all embedded
subobject In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theor ...
s of a given object, up to isomorphism, should also be small, and thus an ordered set. 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 operator). In a
concrete category In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, ''see Relative concreteness below''). This functor makes it possible to think of the objects of ...
, an embedding is a morphism $f:A\rightarrow B$ 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 $f$ is a morphism $fg:C\rightarrow B$, then $g$ itself is a morphism. A factorization system for a category also gives rise to a notion of embedding. If $\left(E,M\right)$ 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 concept, known as quotient. All the preceding properties can be dualized. An embedding can also refer to an embedding functor.

* Closed immersion * Cover *
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 lemma *
Submanifold In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
* Subspace * Universal space

# References

* * * * * * * * * * * * . * * .