dual of Stone representation theorem
The Stone representation theorem characterizes a Boolean algebra as a field of sets in a topological space. There is also a dual to this famous theorem that characterizes a Boolean space as a topological space constructed from a Boolean algebra.
Next, define a function by
Our ultimate goal is to prove that is the desired homeomorphism. We break down the proof of this into several stages:
Pick any maximal ideal of . We want to find an such that . If no such exists, then for every , there is some clopen set such that . This implies that . Since is compact, for some finite set . Since is an ideal, and is a finite join of elements of , we see that . But this would mean that , contradicting the fact that is a maximal, hence a proper ideal of . ∎
and are continuous.
We use a fact about continuous functions between two Boolean spaces:
a bijection is a homeomorphism iff it maps clopen sets to clopen sets (proof here (http://planetmath.org/HomeomorphismBetweenBooleanSpaces)).
So suppose that is clopen in , we want to prove that is clopen in . In other words, there is an element (so that is clopen in ) such that
This is because every clopen set in has the form for some (see the lemma in this entry (http://planetmath.org/StoneRepresentationTheorem)). Now, , the last equality is based on the fact that is a bijection. Thus by setting completes the proof of the lemma. ∎
Therefore, is a homemorphism, and the proof of theorem is complete. ∎
|Title||dual of Stone representation theorem|
|Date of creation||2013-03-22 19:08:38|
|Last modified on||2013-03-22 19:08:38|
|Last modified by||CWoo (3771)|