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| {{Redirect|Isometric embedding|related concepts for [[metric space]]s|isometry}}
| | 30 year old Early Childhood (Pre-Primary School) Teacher Coto from Chalk River, has many hobbies and interests that include entertaining, hay day and rowing. Keeps a tour blog and has plenty to write about after going to Su Nuraxi di Barumini.<br><br>my site [http://i-showad.com/xe/?document_srl=1876212 hay day cheats] |
| {{Other uses}}
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| In [[mathematics]], an '''embedding''' (or '''imbedding''') is one instance of some [[mathematical structure]] contained within another instance, such as a [[group (mathematics)|group]] that is a [[subgroup]].
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| When some object ''X'' is said to be embedded in another object ''Y'', the embedding is given by some [[injective]] and structure-preserving map {{nowrap|''f'' : ''X'' → ''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]], a structure-preserving map is called a [[morphism]].
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| The fact that a map {{nowrap|''f'' : ''X'' → ''Y''}} is an embedding is often indicated by the use of a "hooked arrow", thus: <math> f : X \hookrightarrow Y.</math> On the other hand, this notation is sometimes reserved for [[inclusion map]]s.
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| 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]]s in the [[integer]]s, the integers in the [[rational number]]s, the rational numbers in the [[real number]]s, and the real numbers in the [[complex number]]s. In such cases it is common to identify the [[Domain (mathematics)|domain]] ''X'' with its [[image (mathematics)|image]] ''f''(''X'') contained in ''Y'', so that then {{nowrap|''X'' ⊆ ''Y''}}.
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| ==Topology and geometry==
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| ===General topology===
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| In [[general topology]], an embedding is a [[homeomorphism]] onto its image.<ref>{{citation| first = R.W. | last = Sharpe | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer-Verlag, New York | year = 1997| isbn = 0-387-94732-9}}, page 16.
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| </ref> More explicitly, an [[injective]] [[continuous function (topology)|continuous]] map ''f'' : ''X'' → ''Y'' between [[topological space]]s ''X'' and ''Y'' is a '''topological embedding''' if ''f'' yields a homeomorphism between ''X'' and ''f''(''X'') (where ''f''(''X'') carries the [[topological subspace|subspace topology]] inherited from ''Y''). Intuitively then, the embedding ''f'' : ''X'' → ''Y'' lets us treat ''X'' as a [[topological subspace|subspace]] of ''Y''. Every embedding is [[injective]] and [[continuous function (topology)|continuous]]. Every map that is injective, continuous and either [[open map|open]] or [[closed map|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''.
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| 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 while the other is not.
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| ===Differential topology===
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| In [[differential topology]]:
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| Let ''M'' and ''N'' be smooth [[manifold]]s and <math>f:M\to N</math> be a smooth map. Then ''f'' is called an [[immersion (mathematics)|immersion]] if its [[pushforward (differential)|derivative]] 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]] onto its image).<ref>{{citation| first = F.W. | last = Warner | title = Foundations of Differentiable Manifolds and Lie Groups | publisher = Springer-Verlag, New York | year = 1983| isbn = 0-387-90894-3}}, page 22.</ref>
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| In other words, an embedding is [[diffeomorphism|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 <math>x\in M</math> there is a neighborhood <math>x\in U\subset M</math> such that <math>f:U\to N</math> is an embedding.)
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| When the domain manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion.
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| An important case is ''N''='''R'''<sup>n</sup>. The interest here is in how large ''n'' must be, in terms of the dimension ''m'' of ''M''. The [[Whitney embedding theorem]] <ref>Whitney H., ''Differentiable manifolds,'' Ann. of Math. (2), '''37''' (1936), 645-680.</ref> states that ''n'' = 2''m'' is enough, and is the best possible linear bound. For example the [[real projective plane]] of dimension m requires ''n'' = 2m for an embedding. An immersion of this surface is, however, possible in '''R'''<sup>3</sup>, and one example is [[Boy's surface]]—which has self-intersections. The [[Roman surface]] fails to be an immersion as it contains [[cross-cap]]s.
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| An embedding is '''proper''' if it behaves well [[List_of_mathematical_abbreviations|w.r.t.]] [[Topological_manifold#Manifolds_with_boundary|boundaries]]: one requires the map <math>f: X \rightarrow Y</math> to be such that
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| *<math>f(\partial X) = f(X) \cap \partial Y</math>, and
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| *<math>f(X)</math> is [[Transversality (mathematics)|transversal]] to <math>\partial Y</math> in any point of <math>f(\partial X)</math>.
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| The first condition is equivalent to having <math>f(\partial X) \subseteq \partial Y</math> and <math>f(X \setminus \partial X) \subseteq Y \setminus \partial Y</math>. The second condition, roughly speaking, says that f(X) is not tangent to the boundary of Y.
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| ===Riemannian geometry===
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| In [[Riemannian geometry]]:
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| Let (''M,g'') and (''N,h'') be [[Riemannian manifold]]s.
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| An '''isometric embedding''' is a smooth embedding ''f'' : ''M'' → ''N'' which preserves the [[Riemannian metric|metric]] in the sense that ''g'' is equal to the [[pullback (differential geometry)|pullback]] of ''h'' by ''f'', i.e. ''g'' = ''f''*''h''. Explicitly, for any two tangent vectors
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| :<math>v,w\in T_x(M)</math>
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| we have
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| :<math>g(v,w)=h(df(v),df(w)).\,</math>
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| Analogously, '''isometric immersion''' is an immersion between Riemannian manifolds which preserves the Riemannian metrics.
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| Equivalently, an isometric embedding (immersion) is a smooth embedding (immersion) which preserves length of [[curve]]s (cf. [[Nash embedding theorem]]).<ref>Nash J., ''The embedding problem for Riemannian manifolds,'' Ann. of Math. (2), '''63''' (1956), 20-63.</ref>
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| ==Algebra==
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| In general, for an algebraic category ''C'', an embedding between two ''C''-algebraic structures ''X'' and ''Y'' is a ''C''-morphism ''e:X→Y'' which is injective.
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| ===Field theory===
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| In [[field theory (mathematics)|field theory]], an '''embedding''' of a [[field (mathematics)|field]] ''E'' in a field ''F'' is a [[ring homomorphism]] σ : ''E'' → ''F''.
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| The [[Kernel (algebra)|kernel]] of σ is an [[ideal (ring theory)|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]]. Hence, ''E'' is [[isomorphic]] to the subfield σ(''E'') of ''F''. This justifies the name ''embedding'' for an arbitrary homomorphism of fields.
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| ===Universal algebra and model theory===
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| {{further2|[[Substructure]] and [[Elementary equivalence]]}}
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| If σ is a [[signature (logic)|signature]] and <math>A,B</math> are σ-[[structure (mathematical logic)|structures]] (also called σ-algebras in [[universal algebra]] or models in [[model theory]]), then a map <math>h:A \to B</math> is a σ-embedding [[iff]] all the following holds:
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| * <math>h</math> is [[injective]],
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| * for every <math>n</math>-ary function symbol <math>f \in\sigma</math> and <math>a_1,\ldots,a_n \in A^n,</math> we have <math>h(f^A(a_1,\ldots,a_n))=f^B(h(a_1),\ldots,h(a_n))</math>,
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| * for every <math>n</math>-ary relation symbol <math>R \in\sigma</math> and <math>a_1,\ldots,a_n \in A^n,</math> we have <math>A \models R(a_1,\ldots,a_n)</math> iff <math>B \models R(h(a_1),\ldots,h(a_n)).</math>
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| Here <math>A\models R (a_1,\ldots,a_n)</math> is a model theoretical notation equivalent to <math>(a_1,\ldots,a_n)\in R^A</math>. In model theory there is also a stronger notion of [[elementary embedding]].
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| ==Order theory and domain theory==
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| In [[order theory]], an embedding of [[partial order]]s is a function F from X to Y such that:
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| :<math>\forall x_1,x_2\in X: x_1\leq x_2\Leftrightarrow F(x_1)\leq F(x_2)</math>.
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| In [[domain theory]], an additional requirement is:
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| :<math> \forall y\in Y:\{x: F(x)\leq y\}</math> is [[Directed set|directed]].
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| ==Metric spaces==
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| A mapping <math>\phi: X \to Y</math> of [[metric spaces]] is called an ''embedding''
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| (with distortion <math>C>0</math>) if
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| :<math> L d_X(x, y) \leq d_Y(\phi(x), \phi(y)) \leq CLd_X(x,y) </math>
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| for some constant <math>L>0</math>.
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| === Normed spaces ===
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| An important special case is that of [[normed spaces]]; in this case it is natural to consider linear embeddings.
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| One of the basic questions that can be asked about a finite-dimensional [[normed space]] <math>(X, \| \cdot \|)</math> is, ''what is the maximal dimension <math>k</math> such that the [[Hilbert space]] <math>\ell_2^k</math> can be linearly embedded into <math>X</math> with constant distortion?''
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| The answer is given by [[Dvoretzky's theorem]].
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| ==Category theory== | |
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| In [[category theory]], 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 [[monomorphism|extremal monomorphism]] is an embedding and embeddings are stable under [[Pullback (category theory)|pullback]]s.
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| Ideally the class of all embedded [[subobject]]s of a given object, up to isomorphism, should also be [[small class|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 to define new local structures on the category (such as a [[closure operator]]).
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| In a [[concrete category]], 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:
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| 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.
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| 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.
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| As usual in category theory, there is a [[dual (category theory)|dual]] concept, known as quotient. All the preceding properties can be dualized.
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| An embedding can also refer to an [[Subcategory#Embeddings|embedding functor]].
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| ==See also==
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| *[[Cover (algebra)|Cover]]
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| *[[Immersion (mathematics)|Immersion]]
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| *[[Submanifold]]
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| *[[Subspace (topology)|Subspace]]
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| *[[Closed immersion]]
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| *[[Johnson–Lindenstrauss lemma]]
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| *[[Dimension reduction]]
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| ==Notes==
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| {{reflist}}
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| == References ==
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| * {{citation| first = R.W. | last = Sharpe | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer-Verlag, New York | year = 1997| isbn = 0-387-94732-9}}.
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| * {{citation| first = F.W. | last = Warner | title = Foundations of Differentiable Manifolds and Lie Groups | publisher = Springer-Verlag, New York | year = 1983| isbn = 0-387-90894-3}}.
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| == External links ==
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| *{{cite book|last=Adámek|first=Jiří|coauthors=Horst Herrlich, George Strecker|title=Abstract and Concrete Categories (The Joy of Cats)|url=http://katmat.math.uni-bremen.de/acc/|year=2006}}
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| * [http://www.map.mpim-bonn.mpg.de/Embedding Embedding of manifolds] on the Manifold Atlas
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| [[Category:Abstract algebra]]
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| [[Category:Category theory]]
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| [[Category:General topology]]
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| [[Category:Differential topology]]
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| [[Category:Functions and mappings]]
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| [[Category:Maps of manifolds]]
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| [[Category:Model theory]]
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| [[Category:Order theory]]
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