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)∣a∈B}, |
where M(a)={M∈B*∣a∉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=⋃{M(ai)∣i∈I} for some index set I, since U is open. But U is closed, so B*-U=⋃{M(aj)∣j∈J} for some index set J. Hence B*=⋃{M(ak)∣k∈I∪J}. Since B* is compact, there is a finite subset K of I∪J such that B*=⋃{M(ak)∣k∈K}. Let V=⋃{M(ai)∣i∈K∩I}. Then V⊆U. But B*-V⊆B*-U also. So U=V. Let y=⋁{ai∣i∈K∩I}, which exists because K∩I is finite. As a result,
U=V=⋃{M(ai)∣i∈K∩I}=M(⋁{ai∣i∈K∩I})=M(y)∈ℬ. |
∎
Finally, based on the result of this entry (http://planetmath.org/RepresentingABooleanLatticeByFieldOfSets), B is isomorphic to the field of sets
F:={F(a)∣a∈B}, |
where F(a)={P∣P prime in B, and a∉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 |