dual space of a Boolean algebra
Definition. For any , define , and .
It is know that in a Boolean algebra, maximal ideals and prime ideals coincide. From this entry (http://planetmath.org/RepresentingABooleanLatticeByFieldOfSets), we have the three following properties concerning :
Furthermore, if , then .
From these properties, we see that and . As a result, we see that
is a topological space, whose topology is generated by the basis .
We may in fact treat as a subbasis for , since finite intersections of elements of remain in .
Each is open, by definition, and closed, since it is the complement of the open set . ∎
If such that , then there is some such that and . This means that and , which means that . Since and are open and disjoint, with and , we see that is Hausdorff. ∎
Now, based on a topological fact, every zero-dimensional Hausdorff space is totally disconnected. Hence is totally disconnected.
Suppose is a collection of open sets whose union is . Since each is a union of elements of , we might as well assume that is covered by elements of . In other words, we may assume that each is some .
Let be the ideal generated by the set . If , then can be extended to a maximal ideal . Since each , we see that , so that , which is a contradiction. Therefore, . In particular, , which means that can be expressed as the join of a finite number of the ’s:
where is a finite subset of . As a result, we have
So has a finite subcover, and hence is compact. ∎
Collecting the last three results, we see that is a Boolean space.
|Title||dual space of a Boolean algebra|
|Date of creation||2013-03-22 19:08:35|
|Last modified on||2013-03-22 19:08:35|
|Last modified by||CWoo (3771)|