Embedding

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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 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 : XY. 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 : XY is an embedding is often indicated by the use of a "hooked arrow", thus:  f : X \hookrightarrow Y.

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 then XY.

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Topology and geometry

General topology

In general topology, an embedding is a homeomorphism onto its image. More explicitly, a map f : XY between topological spaces X and Y is an embedding if f yields a homeomorphism between X and f(X) (where f(X) carries the subspace topology inherited from Y). Intuitively then, the embedding f : XY 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(X) is neither an open set nor a closed set in Y.

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