4.2 Statement and proof of Stone's representation theorem

Stone's Representation Theorem links Boolean algebras to fields of sets. It shows how abstract algebraic structures map onto concrete set operations, bridging the gap between algebra and topology.

The theorem proves that every Boolean algebra is isomorphic to a field of sets. This powerful result allows us to visualize and work with Boolean algebras using familiar set operations, making abstract concepts more tangible.

Stone's Representation Theorem

Stone's representation theorem

• Establishes isomorphism between Boolean algebras and fields of sets
• Field of sets comprises clopen subsets of Stone space
• Boolean algebra features operations meet, join, and complement
• Stone space characterized as totally disconnected, compact Hausdorff topological space
• Clopen sets simultaneously closed and open in topology
• Isomorphism preserves Boolean operations and creates bijective mapping

Proof steps for Stone's theorem

1. Construct Stone space using ultrafilters as points
2. Define topology based on principal ultrafilters
3. Prove total disconnectedness by separating points with clopen sets
4. Demonstrate compactness using Alexander's subbase theorem
5. Verify Hausdorff property with disjoint neighborhoods for distinct points
6. Map Boolean algebra elements to clopen sets in Stone space
7. Confirm isomorphism preserves operations and exhibits bijectivity

Constructing and Verifying the Isomorphism

Boolean algebra from Stone space

• Begin with Stone space $X$
• Form set $B$ of all clopen subsets of $X$
• Define operations on $B$: meet (intersection), join (union), complement (relative to $X$)
• Validate Boolean algebra axioms for $B$:
  • Commutativity (order irrelevance)
  • Associativity (grouping irrelevance)
  • Distributivity (distribution over operations)
  • Identity elements (empty set, whole space)
  • Complement laws (inverse relationships)

Isomorphism of constructed algebra

• Create mapping $f$ from original algebra $A$ to $B$
• Define $f(a)$ as set of ultrafilters containing $a$
• Prove homomorphism properties:
  • $f(a \wedge b) = f(a) \cap f(b)$
  • $f(a \vee b) = f(a) \cup f(b)$
  • $f(\neg a) = X \setminus f(a)$
• Demonstrate injectivity using distinct ultrafilter membership
• Show surjectivity by mapping all clopen subsets to $A$ elements
• Conclude $f$ establishes isomorphism between $A$ and $B$