View source for Manifold is not union of images of manifolds of smaller dimension

{{applicationof|Sard's theorem}}

{{topological constraint in smooth category}}

==Statement==

Let <math>M</math> be a [[smooth manifold]] and <math>N_1, N_2, \ldots</math> be a countable sequence of smooth manifolds, each having dimension strictly less than that of <math>M</math>. Suppose <math>f_i:N_i \to M</math> are [[smooth map]]s. Then, <math>M</math> is ''not'' the union of <math>f_i(N_i)</math>.

The analogous statement is not true for [[topological manifold]]s with [[continuous map]]s: there can be a continuous surjective map from a manifold of smaller dimension to a manifold of larger dimension.

==Proof==