A topological characterization of indecomposable sets of finite perimeter

Abstract

We prove that a set of finite perimeter is indecomposable if and only if it is, up to a choice of suitable representative, connected in the 1-fine topology. This gives a topological characterization of indecomposability which is new even in Euclidean spaces. Our approach relies crucially on the metric space theory of functions of bounded variation, and we are able to prove our main result in a complete, doubling metric measure space supporting a $1$-Poincar\'{e} inequality and having the two-sidedness property (this class includes all Riemannian manifolds, Carnot groups, and ${\sf RCD}(K,N)$ spaces with $K\in\mathbb R$ and $N<\infty$). As an immediate corollary, we obtain an alternative proof of the decomposition theorem for sets of finite perimeter into maximal indecomposable components.