Is the Stone space of every finite Boolean algebra a finite discrete space (for every finite Boolean algebra is complete, atomic, and isomorphic to the power set of its atoms; and finite discrete spaces are zero-dimensional, compact, and Hausdorff)?
1 Answer
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The Stone space of a finite Boolean algebra is clearly finite. But every finite Hausdorff space is discrete, hence yes: the Stone space of any finite Boolean algebra must be discrete. Even more is true: the category of finite Stone spaces is isomorphic to the category of finite sets, and there is an (anti-)equivalence between the category of finite sets and the category of finite Boolean algebras (see Johnstone's monograph).
