Stone's representation theorem for Boolean algebras

Remarks

The map $a \mapsto \hat a$ preserves the operations because an ultrafilter contains $a \wedge b$ exactly when it contains $a$ and $b$, and contains $\neg a$ exactly when it does not contain $a$. It is injective by the ultrafilter lemma: if $a \ne b$, then $a \wedge \neg b \ne 0$ or $\neg a \wedge b \ne 0$, and an ultrafilter containing that element separates $a$ and $b$. Compactness of $S(B)$ also uses the ultrafilter lemma, and surjectivity onto the clopen sets follows from compactness, since a clopen set is a finite union of basic sets $\hat a$. The ultrafilter lemma cannot be dropped: over ZF, the theorem implies it.

Example: let $B$ be the algebra of finite and cofinite subsets of $\N$. Its ultrafilters are the principal ones and one more, made of the cofinite sets, and $S(B)$ is the one-point compactification of $\N$. For a finite Boolean algebra the space is finite and discrete, which gives Stone's representation theorem for finite Boolean algebras.

Going deeper

For a student

The theorem is the object part of Stone duality, which also matches homomorphisms with continuous maps. It implies that an equation holds in all Boolean algebras as soon as it holds in the two-element algebra. This is the completeness theorem for classical propositional logic in algebraic form. The same idea, with prime filters instead of ultrafilters, gives Priestley duality for distributive lattices.