An embedding is a representation of a topological object, manifold, graph, field, etc. in a certain space in such a way that its connectivity or algebraic properties are preserved. For example, a field embedding preserves the algebraic structure of plus and times, while an embedding of a topological space preserves open sets. In a graph embedding in a surface, distinct vertices are represented by distinct points and edges by arcs whose only intersections are common endpoints.
Geometric graph terminology also uses the word embedding for some drawings defined by coordinate or distance constraints. A unit-distance embedding, for example, places distinct vertices at distinct points and represents edges by line segments of unit length without requiring the line segments to be disjoint. When two of these line segments intersect away from a common endpoint, the result is a geometric drawing rather than a topological embedding.
One space 
is embedded in another space 
when the properties of 
restricted to 
are the same as the properties of 
. For example, the rationals are embedded in the reals, and
the integers are embedded in the rationals. In geometry, the sphere is embedded in
as the unit sphere.
Let 
and 
be structures for the same first-order language 
, and let 
be a homomorphism from 
to 
.
Then 
is an embedding provided that it is injective
(Enderton 1972, Grätzer 1979, Burris and Sankappanavar 1981).
For example, if 
and 
are partially ordered sets,
then an injective monotone mapping 
may not be an embedding from 
into 
. To be an embedding, such a mapping must preserve
order "both ways":
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