  We prove topological regularity results for isoperimetric sets in PI spaces
having a suitable deformation property, which prescribes a control on the
increment of the perimeter of sets under perturbations with balls. More
precisely, we prove that isoperimetric sets are open, satisfy boundary density
estimates and, under a uniform lower bound on the volumes of unit balls, are
bounded. Our results apply, in particular, to the class of possibly collapsed
$\mathrm{RCD}(K,N)$ spaces. As a consequence, the rigidity in the isoperimetric
inequality on possibly collapsed $\mathrm{RCD}(0,N)$ spaces with Euclidean
volume growth holds without the additional assumption on the boundedness of
isoperimetric sets. Our strategy is of interest even in the Euclidean setting,
as it simplifies some classical arguments.
