M,N smooth manifolds,dim M=m,dim N=n.m>n.
f:M——>N a smooth map.
d is an integer,0<d<n.
Then B={x∈M: rank Df(x)<d} is a closed set.
please help me,thanks!
f is smooth and the function $\displaystyle x\rightarrow \omega(x)=rank(Df)(x)$ is discrete. So
$\displaystyle B=\{x:rank(Df)(x)<d\}=\{x:rank(DF)(x)=0\}\cup\{x:r ank(Df)(x)=1\}\cup\ldots\cup\{x:rank(Df)(x)=d-1\}=\omega^{-1}(0)\cup\omega^{-1}(1)\cup\ldots\cup \omega^{-1}(d-1)$
is a finite union of closed sets.