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# 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 of $B$, which is composed of all maximal ideals of $B$. According to this entry, $X$ 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, $B$ 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.

## Mathematics Subject Classification

54D99*no label found*06E99

*no label found*03G05

*no label found*

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