  We study a measure-theoretic notion of connectedness for sets of finite
perimeter in the setting of doubling metric measure spaces supporting a weak
$(1,1)$-Poincar\'{e} inequality. The two main results we obtain are a
decomposition theorem into indecomposable sets and a characterisation of
extreme points in the space of BV functions. In both cases, the proof we
propose requires an additional assumption on the space, which is called
isotropicity and concerns the Hausdorff-type representation of the perimeter
measure.
