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":

See also

Campbell's Theorem, Embeddable Knot, Embedded Surface, Extrinsic Curvature, Field, Graph Embedding, Hyperboloid Embedding, Injection, Manifold, Nash's Embedding Theorem, Submanifold

Portions of this entry contributed by Todd Rowland

Portions of this entry contributed by Matt Insall (author's link)

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References

Balzin, E. Lectures for MAA 306 Course: Topics in Differential Geometry. Jun. 7, 2021. https://eduard-balzin.perso.math.cnrs.fr/Differential_geometry_lecture_notes_070621.pdf.Burris, S. and Sankappanavar, H. P. A Course in Universal Algebra. New York: Springer-Verlag, 1981. https://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html.Enderton, H. B. A Mathematical Introduction to Logic. New York: Academic Press, 1972.Grätzer, G. Universal Algebra, 2nd ed. New York: Springer-Verlag, 1979.

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Embedding

Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland and Matt Insall. "Embedding." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Embedding.html

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