# Manifold is not union of images of manifolds of smaller dimension

From Diffgeom

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

*This article gives a purely topological (or set-theoretic) constraint that we get when we restrict ourselves to the category of differential manifolds with smooth maps*

## Statement

Let be a smooth manifold and be a countable sequence of smooth manifolds, each having dimension strictly less than that of . Suppose are smooth maps. Then, is *not* the union of .

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.