  Let $(M^n_i, g_i)\to (X,d_X)$ be a Gromov-Hausdorff converging sequence of
Riemannian manifolds with ${\rm Sec}_{g_i} \ge -1$, ${\rm diam}\, (M_i)\le D$,
and such that the $M^n_i$ are all homeomorphic to tori $T^n$. Then $X$ is
homeomorphic to a $k$-dimensional torus $T^k$ for some $0\leq k\leq n$. This
answers a question of Petrunin in the affirmative. We show this result is false
is the $M^n_i$ are homeomorphic tori which are only assumed to be Alexandrov
spaces. When $n=3$, we prove the same tori stability under the weaker condition
${\rm Ric}_{g_i} \ge -2$.
