# What are some theorems made easier by Stone Duality?

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.

Thanks!

• The free Boolean algebra on $$n$$ elements has $$2^{2^n}$$ elements. (Dually, an $$n$$-tuple of clopen subsets of a Stone space is just a map to $$\{0,1\}^n$$, whose clopen algebra has $$2^{2^n}$$ elements.)
• The equational axioms that are true in all Boolean algebras are exactly those that are true in $$\{0,1\}$$, i.e. the propositional tautologies. (Immediate from the fact that every Boolean algebra embeds in a power of $$\{0,1\}$$, namely the power set of its Stone space.)
• Every nontrivial countable Boolean algebra is a retract of the free Boolean algebra on countably many generators. (Dually, we want to show every nonempty closed subspace of the Cantor set is a retract. This is easy by a geometric argument; e.g., considering the usual Cantor set as a subset of $$[0,1]$$ and a nonempty closed subspace $$X$$, map each point of the Cantor set to the nearest point in $$X$$, with a little care at the endpoints where it's possible to have a tie.)