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

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 |