Embedding

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. 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 → 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. The fact that a map f : X → Y is an embedding is often indicated by the use of a 'hooked arrow' (.mw-parser-output .monospaced{font-family:monospace,monospace}U+21AA ↪ .mw-parser-output .smallcaps{font-variant:small-caps}RIGHTWARDS ARROW WITH HOOK); thus: f : X ↪ Y . {displaystyle f:Xhookrightarrow 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 numbers in the integers, the integers in the rational numbers, the rational numbers in the real numbers, and the real numbers in the complex numbers. In such cases it is common to identify the domain X with its image f(X) contained in Y, so that X ⊆ Y. In general topology, an embedding is a homeomorphism onto its image. More explicitly, an injective continuous map f : X → Y {displaystyle f:X o Y} between topological spaces X {displaystyle X} and Y {displaystyle Y} is a topological embedding if f {displaystyle f} yields a homeomorphism between X {displaystyle X} and f ( X ) {displaystyle f(X)} (where f ( X ) {displaystyle f(X)} carries the subspace topology inherited from Y {displaystyle Y} ). Intuitively then, the embedding f : X → Y {displaystyle f:X o Y} lets us treat X {displaystyle X} as a subspace of Y {displaystyle 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 ) {displaystyle f(X)} is neither an open set nor a closed set in Y {displaystyle Y} . For a given space Y {displaystyle Y} , the existence of an embedding X → Y {displaystyle X o Y} is a topological invariant of X {displaystyle X} . This allows two spaces to be distinguished if one is able to be embedded in a space while the other is not. In differential topology:Let M {displaystyle M} and N {displaystyle N} be smooth manifolds and f : M → N {displaystyle f:M o N} be a smooth map. Then f {displaystyle f} is called an immersion if its 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). 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. An immersion is a local embedding (i.e. for any point x ∈ M {displaystyle xin M} there is a neighborhood x ∈ U ⊂ M {displaystyle xin Usubset M} such that f : U → N {displaystyle f:U o N} is an embedding.)