embeddability

Embeddability is a concept in mathematics that concerns representing an abstract object within a larger ambient space in a way that preserves its structure. In topology and geometry, a space X is embeddable into a space Y if there exists an embedding of X into Y. An embedding is an injective, continuous map that is a homeomorphism between X and its image with the subspace topology inherited from Y. Embeddability thus requires both a faithful mapping and the preservation of topological structure.

A standard case is topological embedding, where one asks whether a given space can be realized as a subset of a Euclidean space. For example, the circle S1 embeds in the plane R2 in the usual way, and the n-sphere Sn embeds in Rn+1. Some spaces require higher ambient dimensions to embed; for instance, the real projective plane RP2 cannot embed in R3 but does embed in R4.

Key results help delineate embeddability limitations. Whitney’s embedding theorem states that every smooth m-dimensional manifold can be embedded in Euclidean space R2m. A stronger version (the Whitney immersion theorem) concerns immersions, while Nash’s embedding theorem guarantees that every smooth Riemannian manifold can be isometrically embedded into some Euclidean space. In graph theory, embeddability concerns drawing graphs on surfaces without edge crossings; planar graphs are those embeddable in the plane, while more complex graphs may require surfaces of higher genus.

Applications of embeddability span geometry, topology, and computer science, including questions about dimension, visualization, and the representation of abstract structures within concrete spaces.