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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!

1 Answers1

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Here are a few simple examples of theorems that are made more obvious using Stone duality. All of these also have not-too-hard proofs without it, but I think that is inevitable, since Stone duality itself is not too hard to prove.

  • Every finite Boolean algebra is isomorphic to a power set. (Obvious since the dual space is finite, and trivially a finite Stone space is discrete.)
  • Every infinite Boolean algebra has infinitely many ultrafilters. (If it had finitely many, the Stone space would be finite, so it would have only finitely many subsets.)
  • 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.)
  • The category of Stone spaces is complete and cocomplete. (The category of Boolean algebras obviously is by general algebraic considerations. Or alternatively, you can go the other direction: the fact that Stone spaces have limits is pretty obvious (products are just the topological product and equalizers are just taking the equalizer as sets as a subspace), so you can deduce that Boolean algebras have colimits.)
  • 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.)
  • The category of Stone spaces is the pro-completion of the category of finite sets. (The dual statement is that the category of Boolean algebras is the ind-completion of the opposite category of finite sets, which is just the category of finite Boolean algebras by Stone duality for finite discrete spaces. This is then obvious since Boolean algebras are finitary algebraic structures and every finitely generated Boolean algebra is finite.)
Eric Wofsey
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