User:Dfeuer/Stone's Representation Theorem for Boolean Algebras

Theorem
If $B$ is a Boolean algebra and $S(B)$ is the Stone space of $B$, then

$B$ is isomorphic to the algebra of clopen subsets of $S(B)$, where the isomorphism sends an element $b \in B$ to the set of all ultrafilters that contain $b$.

Proof
For each $b \in B$, let $E\left({b}\right)$ be the set of ultrafilters in $B$ containing $b$.

Lemma
If $b \in B$, then $E\left({b}\right)$ is clopen in $S(B)$.

Proof
$E(b)$ and $E(\neg b)$ are open by the definition of the Stone Space.

Then since an ultrafilter on a Boolean algebra contains either an element or its complement but not both,
 * $E(b) \cup E(\neg b) = S(B)$

and
 * $E(b)\cap E(\neg b) = \varnothing$.