  For a positive integer $k$, a group $G$ is said to be totally $k$-closed if
in each of its faithful permutation representations, say on a set $\Omega$, $G$
is the largest subgroup of $\operatorname{Sym}(\Omega)$ which leaves invariant
each of the $G$-orbits in the induced action on $\Omega\times\dots\times
\Omega=\Omega^k$. We prove that every abelian group $G$ is totally
$(n(G)+1)$-closed, but is not totally $n(G)$-closed, where $n(G)$ is the number
of invariant factors in the invariant factor decomposition of $G$. In
particular, we prove that for each $k\geq2$ and each prime $p$, there are
infinitely many finite abelian $p$-groups which are totally $k$-closed but not
totally $(k-1)$-closed. This result in the special case $k=2$ is due to
Abdollahi and Arezoomand. We pose several open questions about total
$k$-closure.
