Sard’s theorem

Let $\varphi :X^n\to Y^m$ be a smooth map on smooth manifolds. A critical point of $\varphi$ is a point $p\in X$ such that the differential ${\varphi }_{*}:T_{p}X\to T_{\varphi (p)}Y$ considered as a linear transformation of real vector spaces has rank (http://planetmath.org/RankLinearMapping) . A critical value of $\varphi$ is the image of a critical point. A regular value of $\varphi$ is a point $q\in Y$ which is not the image of any critical point. In particular, $q$ is a regular value of $\varphi$ if $q\in Y\setminus \varphi (X)$.

Following Spivak [Spivak], we say a subset $V$ of $Y^m$ has measure zero if there is a sequence of coordinate charts $(x_i,U_i)$ whose union contains $V$ and such that $x_i(U_i\cap V)$ has measure 0 (in the usual sense) in ${ℝ}^m$ for all $i$. With that definition, we can now state:

Sard’s Theorem. Let $\varphi :X\to Y$ be a smooth map on smooth manifolds. Then the set of critical values of $\varphi$ has measure zero.