  Let $m$ be a positive integer and let $\Omega$ be a finite set. The
$m$-closure of $G\le$Sym$(\Omega)$ is the largest permutation group on $\Omega$
having the same orbits as $G$ in its induced action on the Cartesian product
$\Omega^m$. The exact formula for the $m$-closure of the wreath product in
product action is given. As a corollary, a sufficient condition is obtained for
this $m$-closure to be included in the wreath product of the $m$-closures of
the factors.
