  This paper studies regularity of perimiter quasiminimizing sets in metric
measure spaces with a doubling measure and a Poincare inequality. The main
result shows that the measure theoretic boundary of a quasiminimizing set
coincides with the topological boundary. We also show that such a set has
finite Minkowski content and apply the regularity theory to study
rectifiability issues related to quasiminimal sets in strong
A_{\infty}-weighted Euclidean case.
