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.

When some object

X

{\displaystyle X}

is said to be embedded in another object

Y

{\displaystyle Y}

, the embedding is given by some injective and structure-preserving map

f

:

X

→

Y

{\displaystyle f:X\rightarrow Y}

. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which

X

{\displaystyle X}

and

Y

{\displaystyle 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

{\displaystyle f:X\rightarrow Y}

is an embedding is often indicated by the use of a "hooked arrow" (U+21AA ↪ RIGHTWARDS ARROW WITH HOOK); thus:

f

:

X

↪

Y

.

{\displaystyle f:X\hookrightarrow Y.}

(On the other hand, this notation is sometimes reserved for inclusion maps.)

Given

X

{\displaystyle X}

and

Y

{\displaystyle Y}

, several different embeddings of

X

{\displaystyle X}

in

Y

{\displaystyle 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

{\displaystyle X}

with its image

f

(

X

)

{\displaystyle f(X)}

contained in

Y

{\displaystyle Y}

, so that

X

⊆

Y

{\displaystyle X\subseteq Y}

.

Topology and geometry

General topology

In general topology, an embedding is a homeomorphism onto its image. More explicitly, an injective continuous map

f

:

X

→

Y

{\displaystyle f:X\to 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}

).