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.
For topological spaces, an embedding 
is a homeomorphism
from 
onto its image 
with the subspace
topology inherited from 
.
An injective continuous
map need not satisfy this condition. A smooth embedding between smooth
manifolds is an immersion that is also an embedding
in this topological sense (Balzin 2021, pp. 44-46). For example, 
, 
, is a homeomorphism,
but its derivative vanishes at zero, so it is not
a smooth embedding.
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 rational
numbers are embedded in the real numbers, and
the integers are embedded in the rational
numbers. 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
and reflects every relation: for each 
-ary relation symbol 
, 
holds iff 
holds (Enderton 1972, Grätzer 1979,
Burris and Sankappanavar 1981).
For example, if 
and 
are partially
ordered sets, then an injective monotone map 
may not be an embedding from
into 
. To be an embedding, such a map
must preserve order "both ways":
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