# M. H. Stone’s representation theorem

###### Theorem 1.

Given a Boolean algebra $B$ there exists a totally disconnected compact Hausdorff space $X$ such that $B$ is isomorphic to the Boolean algebra of clopen subsets of $X$.

###### Proof.

Let $X=B^{*}$, the dual space (http://planetmath.org/DualSpaceOfABooleanAlgebra) of $B$, which is composed of all maximal ideals of $B$. According to this entry (http://planetmath.org/DualSpaceOfABooleanAlgebra), $X$ is a Boolean space (totally disconnected compact Hausdorff) whose topology is generated by the basis

 $\mathcal{B}:=\{M(a)\mid a\in B\},$

where $M(a)=\{M\in B^{*}\mid a\notin M\}$.

Next, we show a general fact about the dual space $B^{*}$:

###### Lemma 2.

$\mathcal{B}$ is the set of all clopen sets in $X$.

###### Proof.

Clearly, every element of $\mathcal{B}$ is clopen, by definition. Conversely, suppose $U$ is clopen. Then $U=\bigcup\{M(a_{i})\mid i\in I\}$ for some index set $I$, since $U$ is open. But $U$ is closed, so $B^{*}-U=\bigcup\{M(a_{j})\mid j\in J\}$ for some index set $J$. Hence $B^{*}=\bigcup\{M(a_{k})\mid k\in I\cup J\}$. Since $B^{*}$ is compact, there is a finite subset $K$ of $I\cup J$ such that $B^{*}=\bigcup\{M(a_{k})\mid k\in K\}$. Let $V=\bigcup\{M(a_{i})\mid i\in K\cap I\}$. Then $V\subseteq U$. But $B^{*}-V\subseteq B^{*}-U$ also. So $U=V$. Let $y=\bigvee\{a_{i}\mid i\in K\cap I\}$, which exists because $K\cap I$ is finite. As a result,

 $U=V=\bigcup\{M(a_{i})\mid i\in K\cap I\}=M(\bigvee\{a_{i}\mid i\in K\cap I\})=% M(y)\in\mathcal{B}.$

Finally, based on the result of this entry (http://planetmath.org/RepresentingABooleanLatticeByFieldOfSets), $B$ is isomorphic to the field of sets

 $F:=\{F(a)\mid a\in B\},$

where $F(a)=\{P\mid P\mbox{ prime in }B,\mbox{ and }a\notin P\}$. Realizing that prime ideals and maximal ideals coincide in any Boolean algebra, the set $F$ is precisely $\mathcal{B}$. ∎

Remark. There is also a dual version of the Stone representation theorem, which says that every Boolean space is homeomorphic to the dual space of some Boolean algebra.

 Title M. H. Stone’s representation theorem Canonical name MHStonesRepresentationTheorem Date of creation 2013-03-22 13:25:34 Last modified on 2013-03-22 13:25:34 Owner rspuzio (6075) Last modified by rspuzio (6075) Numerical id 19 Author rspuzio (6075) Entry type Theorem Classification msc 54D99 Classification msc 06E99 Classification msc 03G05 Synonym Stone representation theorem Synonym Stone’s representation theorem Related topic RepresentingABooleanLatticeByFieldOfSets Related topic DualSpaceOfABooleanAlgebra