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

$$ℬ:=\{M(a)\mid a\in B\},$$

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

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

Lemma 2.

$ℬ$ is the set of all clopen sets in $X$.

Proof.

Clearly, every element of $ℬ$ 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,

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

∎

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

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