In the geometry of manifolds, a manifold *M* given abstractly is considered as a candidate to be embedded in Euclidean space of given dimension *n* (at least dim *M*, naturally: see invariance of domain). That means we look for a submanifold of *n*-dimensional Euclidean space that is at least homeomorphic to *M*. If *M* is smooth we shall want a diffeomorphism; in the case of a Riemannian manifold an isometry (cf. Nash embedding theorem). The interest here is in how large *n* must be, in terms of the dimension *m* of *M*. The basic results of differential topology here concern the case *n* = 2*m*. For example the real projective plane of dimension 2 requires *n* = 4 for an embedding. The less restrictive condition of *immersion* applies to the Boy's surface - which has self-intersections.

In fact the occasion of the proof by Hassler Whitney of the embedding theorem for smooth manifolds is said (rather surprisingly) to have been the first complete exposition of the *manifold concept* (which had been implicit in Riemann's work, Lie group theory, and general relativity for many years); building on Hermann Weyl's book *The Idea of a Riemann surface*.

In domain theory, an **embedding** is a complete partial order F in [X -> Y] is an embedding if

- For all x1, x2 in X, x1 <= x2 <=> F x1 <= F x2 and
- For all y in Y, x | F x <= y is directed.

`\\sqsubseteq`

).
*Based on an article from FOLDOC, used by permission.*