Editing Field of sets (section)

=== Separative and compact fields of sets: towards Stone duality ===

* A field of sets is called '''separative''' (or '''differentiated''') if and only if for every pair of distinct points there is a complex containing one and not the other.
* A field of sets is called '''compact''' if and only if for every proper [[Filter (set theory)|filter]] over <math>X</math> the intersection of all the complexes contained in the filter is non-empty.

These definitions arise from considering the [[topological space|topology]] generated by the complexes of a field of sets. (It is just one of notable topologies on the given set of points; it often happens that another topology is given, with quite different properties, in particular, not zero-dimensional). Given a field of sets <math>\mathbf{X} = ( X, \mathcal{F} )</math> the complexes form a [[Base (topology)|base]] for a topology. We denote by <math>T(\mathbf{X})</math> the corresponding topological space, <math>( X, \mathcal{T} )</math> where <math>\mathcal{T}</math> is the topology formed by taking arbitrary unions of complexes. Then

* <math>T(\mathbf{X})</math> is always a [[zero-dimensional space]].
* <math>T(\mathbf{X})</math> is a [[Hausdorff space]] if and only if <math>\mathbf{X}</math> is separative.
* <math>T(\mathbf{X})</math> is a [[compact space]] with compact open sets <math>\mathcal{F}</math> if and only if <math>\mathbf{X}</math> is compact.
* <math>T(\mathbf{X})</math> is a [[Boolean space]] with [[clopen set]]s <math>\mathcal{F}</math> if and only if <math>\mathbf{X}</math> is both separative and compact (in which case it is described as being '''descriptive''')

The Stone representation of a Boolean algebra is always separative and compact; the corresponding Boolean space is known as the [[Stone space]] of the Boolean algebra. The clopen sets of the Stone space are then precisely the complexes of the Stone representation. The area of mathematics known as [[Stone duality]] is founded on the fact that the Stone representation of a Boolean algebra can be recovered purely from the corresponding Stone space whence a [[Duality (mathematics)|duality]] exists between Boolean algebras and Boolean spaces.