Manifold is not union of images of manifolds of smaller dimension

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This fact is an application of the following pivotal fact/result/idea: Sard's theorem
View other applications of Sard's theorem OR Read a survey article on applying Sard's theorem

Statement

Let M be a smooth manifold and N_1, N_2, \ldots be a countable sequence of smooth manifolds, each having dimension strictly less than that of M. Suppose f_i:N_i \to M are smooth maps. Then, M is not the union of f_i(N_i).

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

Proof