Whitney embedding theorem

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, particularly in <a href="/facts/Differential_topology/Y21eGQe0">differential topology</a>, there are two Whitney embedding theorems, named after <a href="/facts/Hassler_Whitney/ljAyGQME">Hassler Whitney</a>:

<ul><li>The strong Whitney embedding theorem states that any <a href="/facts/Differentiable_manifold/aSnbXKcV">smooth</a> <a href="/facts/Real_numbers/R02gw5Pb">real</a> m-<a href="/facts/Dimension_(mathematics)/0ktmUyac">dimensional</a> <a href="/facts/Manifold/rXPERJtu">manifold</a> (required also to be <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> and <a href="/facts/Second-countable/qw9YdnBN">second-countable</a>) can be <a href="/facts/Smooth_map/mffL4ch2">smoothly</a> <a href="/facts/Embedding/HKeVKc42">embedded</a> in the <a href="/facts/Real_coordinate_space/tEtvw03F">real 2m-space</a>, ⁠
 
 
 
 
 
 R
 
 
 2
 m
 
 
 ,
 
 
 {\displaystyle \mathbb {R} ^{2m},}
 
⁠ if m > 0. This is the best linear bound on the smallest-dimensional Euclidean space that all m-dimensional manifolds embed in, as the <a href="/facts/Real_projective_space/svbABDtd">real projective spaces</a> of dimension m cannot be embedded into real (2m − 1)-space if m is a <a href="/facts/Power_of_two/MA6WEpTt">power of two</a> (as can be seen from a <a href="/facts/Characteristic_class/A6NV9tUu">characteristic class</a> argument, also due to Whitney).</li>
<li>The weak Whitney embedding theorem states that any continuous function from an n-dimensional manifold to an m-dimensional manifold may be approximated by a smooth embedding provided m > 2n. Whitney similarly proved that such a map could be approximated by an <a href="/facts/Immersion_(mathematics)/QMqIgkVY">immersion</a> provided m > 2n − 1. This last result is sometimes called the <a href="/facts/Whitney_immersion_theorem/v0NzQM8v">Whitney immersion theorem</a>.</li></ul>

Whitney embedding theorem open-in-new

Whitney embedding theorem