Theorem

Whitney Embedding Theorem

Topology

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

Statement

The 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

Statement Form

Inequality 1

Connections

Source Whitney embedding theorem (Wikipedia)

Sources

1. Whitney embedding theorem (Wikipedia)

Wikimedia Foundation

Whitney 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:
In Group: Geometric and Differential Topology, Lead section
particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney

Switch Atlas

The Inner World

The Civil World

The Natural World

Whitney Embedding Theorem

Has Statement Form

In Branch

Proved By