Boolean Algebras and Stone Duality
Boolean Spaces and Stone Spaces
The topological spaces that arise as duals of Boolean algebras: compact, Hausdorff, totally disconnected, with a basis of clopen sets.
Learning objectives
- Define Boolean space and verify the defining properties
- Construct the Stone space of a Boolean algebra
- Identify standard examples of Boolean spaces
Boolean spaces
A topological space that is compact, Hausdorff, and totally disconnected — equivalently, compact Hausdorff with a basis of clopen sets. Also called a Stone space or a profinite space.
Total disconnectedness means the connected components are single points. In the presence of compactness and the Hausdorff property this is equivalent to having a basis of sets that are simultaneously closed and open.
| Space | Corresponding Boolean algebra |
|---|---|
| A finite discrete space on n points | The finite Boolean algebra with n atoms |
| The Cantor set | The free countably generated Boolean algebra |
| 2I with the product topology | The free Boolean algebra on I generators |
| βN, the ultrafilters on N | Su(N) |
| The one-point compactification of a discrete space | The finite–cofinite algebra |
| The p-adic integers | A countable Boolean algebra |
Any Boolean space with no isolated points and a countable basis is homeomorphic to the Cantor set. This is the topological counterpart of the uniqueness of the countable atomless Boolean algebra.
The Stone space construction
For a Boolean algebra B, the set of ultrafilters of B, topologised by taking as a basis the sets Nb = {U : b ∈ U} for b ∈ B.
The Stone space of any Boolean algebra is compact, Hausdorff and totally disconnected, and the sets Nb are exactly its clopen subsets.
- The Nb form a basis. Na∧b = Na ∩ Nb, so the family is closed under finite intersection.
- Each Nb is clopen. Its complement is Nb′, because an ultrafilter contains exactly one of b and b′.
- Hausdorff. Distinct ultrafilters differ on some b, and then Nb and Nb′ separate them.
- Compact. A family of basic closed sets with the finite intersection property generates a proper filter, which extends to an ultrafilter by BPI — and that ultrafilter lies in every member of the family.
Step 4 is where the prime ideal theorem enters, and it is unavoidable. Compactness of the Stone space is equivalent to BPI over ZF, which is why Stone duality is a choice-dependent theorem.
Recovering the algebra
The clopen subsets of B* form a Boolean algebra under union, intersection and complement, and the map b ↦ Nb is an isomorphism from B onto it.
Injectivity uses BPI: distinct elements are separated by some ultrafilter. Surjectivity uses compactness: a clopen set is a union of basic sets, and compactness reduces the union to a finite one, whose join is the required element.
Frequently asked questions
Why totally disconnected?
Because clopen sets separate points, and a connected subset containing two points could not be split by a clopen set. Total disconnectedness is exactly what having enough clopen sets amounts to.
Is every compact Hausdorff space a Boolean space?
No. The unit interval is compact Hausdorff but connected, so it has only the two trivial clopen subsets and is far from totally disconnected.
Source. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition — a corrected re-typesetting of Springer-Verlag Graduate Texts in Mathematics 78 (1981). Section IV.4, book pages 152-155.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates and reorganises mathematical results; it is not a reproduction of the source text.