I have seen a lot of praise for the Stone Duality Theorem, which links the algebraic structure of boolean algebras to the topological structure of stone spaces by a (contravariant) adjoint equivalence of categories.
What are some theorems which are made obvious by using duality, or which don't have proofs without duality?
I know that it (and its generalizations) have inspired a lot of work in pointless topology, which looks interesting to me, but it's not what I'm looking for. Ideally these proofs should be theorems about boolean algebras or stone spaces - things which someone could have come up with before the duality was known.
I'm sure these theorems must exist, because Stone Duality, while independently beautiful, is often cited as a useful and powerful result... So I'm not sure why I'm struggling to find witnesses to its utility.