4.1 Stone spaces and Boolean spaces

Stone spaces are a fascinating intersection of topology and algebra. They're compact, totally disconnected Hausdorff spaces with a basis of clopen sets. These spaces are key to understanding the deep connection between Boolean algebras and certain topological spaces.

Stone spaces exemplify the power of duality in mathematics. The Stone representation theorem shows how every Boolean algebra is isomorphic to the algebra of clopen sets of its Stone space, revealing a beautiful correspondence between algebraic and topological structures.

Stone Spaces and Boolean Spaces

Definition of Stone spaces

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• Stone spaces embody topological spaces characterized by total disconnectedness, compactness, and Hausdorff property
• Clopen sets constitute basis for topology allowing every point to have neighborhood basis of clopen sets
• Homeomorphic to closed subspace of Cantor space enables concrete representation
• Stone duality establishes correspondence between Stone spaces and Boolean algebras providing contravariant equivalence of categories

Stone spaces vs Boolean algebras

• Stone representation theorem asserts every Boolean algebra isomorphic to algebra of clopen sets of its Stone space
• Stone functor maps Boolean algebras to Stone spaces while clopen set functor maps Stone spaces to Boolean algebras
• Topological properties mirror algebraic properties (connectedness components correspond to atoms, irreducible closed sets to prime filters)

Examples of Stone spaces

• Cantor space exemplifies prototypical Stone space homeomorphic to $\{0,1\}^\omega$ with product topology
• Finite discrete spaces correspond to finite Boolean algebras
• Stone-Čech compactification of discrete spaces represents Stone space of power set Boolean algebra
• Profinite completions arise from inverse limits of finite discrete spaces

Boolean spaces as Stone spaces

• Boolean spaces defined as zero-dimensional compact Hausdorff spaces with basis of clopen sets
• Proof that Boolean spaces are Stone spaces:
  1. Demonstrate total disconnectedness
  2. Show satisfaction of Stone space definition
  3. Apply Urysohn's lemma to construct continuous functions separating points
• Key steps involve separating distinct points with clopen sets, proving [object Object],[object Object] property, and concluding total disconnectedness
• Every Boolean space possesses Stone dual in Boolean algebra category, establishing equivalence between Stone space and Boolean space categories