Whitney Immersion Theorem

The Whitney Immersion Theorem, named after Hassler Whitney, states that for a dimension m greater than one, any smooth m-dimensional manifold that is Hausdorff and second-countable has a one-to-one immersion into Euclidean 2m-dimensional space, and can be immersed, not necessarily one-to-one, into (2m-1)-dimensional space; every such manifold can likewise be immersed in the (2m-1)-dimensional sphere, which removes the restriction to m greater than one. The weaker version of the result, into (2m+1)-dimensional space, follows simply from a dimension-counting transversality argument, since two m-dimensional manifolds sitting inside 2m-dimensional Euclidean space intersect generically only in a zero-dimensional set.