Any smooth n-dimensional manifold can be smoothly embedded into Euclidean space of dimension 2n. Proved by Hassler Whitney, it shows that abstractly defined manifolds always admit a concrete realization inside ordinary Euclidean space.
Facts
StatementThe strong Whitney embedding theorem states that any smooth real m-dimensional manifold, provided it is Hausdorff and second-countable, can be smoothly embedded into real 2m-dimensional Euclidean space whenever m is greater than zero. The weak Whitney embedding theorem states that a continuous map from an n-dimensional manifold to an m-dimensional manifold can be approximated by a smooth embedding whenever m is greater than 2n. 1 Classification
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Source Whitney embedding theorem (Wikipedia)
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1. Whitney embedding theorem (Wikipedia)
Wikimedia FoundationWhitney embedding theorem, lead section
The strong Whitney embedding theorem states that any smooth real m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the real 2m-space
Proved By: Hassler Whitney, Lead paragraph
In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney:
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