Editing Field of sets (section)

=== Topological fields of sets ===

A '''topological field of sets''' is a triple <math>( X, \mathcal{T}, \mathcal{F} )</math> where <math>( X, \mathcal{T} )</math> is a [[topological space]] and <math>( X, \mathcal{F} )</math> is a field of sets which is closed under the [[closure operator]] of <math>\mathcal{T}</math> or equivalently under the [[interior operator]] i.e. the closure and interior of every complex is also a complex. In other words, <math>\mathcal{F}</math> forms a subalgebra of the power set [[interior algebra]] on <math>( X, \mathcal{T} ).</math>

Topological fields of sets play a fundamental role in the representation theory of interior algebras and [[Heyting algebra]]s. These two classes of algebraic structures provide the [[Algebraic semantics (mathematical logic)|algebraic semantics]] for the [[modal logic]] ''S4'' (a formal mathematical abstraction of [[Epistemic|epistemic logic]]) and [[intuitionistic logic]] respectively. Topological fields of sets representing these algebraic structures provide a related topological [[Semantics of logic|semantics]] for these logics.

Every interior algebra can be represented as a topological field of sets with the underlying Boolean algebra of the interior algebra corresponding to the complexes of the topological field of sets and the interior and closure operators of the interior algebra corresponding to those of the topology. Every [[Heyting algebra]] can be represented by a topological field of sets with the underlying lattice of the Heyting algebra corresponding to the lattice of complexes of the topological field of sets that are open in the topology. Moreover the topological field of sets representing a Heyting algebra may be chosen so that the open complexes generate all the complexes as a Boolean algebra. These related representations provide a well defined mathematical apparatus for studying the relationship between truth modalities (possibly true vs necessarily true, studied in modal logic) and notions of provability and refutability (studied in intuitionistic logic) and is thus deeply connected to the theory of [[modal companion]]s of [[intermediate logic]]s.

Given a topological space the [[Topology glossary|clopen]] sets trivially form a topological field of sets as each clopen set is its own interior and closure. The Stone representation of a Boolean algebra can be regarded as such a topological field of sets, however in general the topology of a topological field of sets can differ from the topology generated by taking arbitrary unions of complexes and in general the complexes of a topological field of sets need not be open or closed in the topology.

==== Algebraic fields of sets and Stone fields ====

A topological field of sets is called '''algebraic''' if and only if there is a base for its topology consisting of complexes.

If a topological field of sets is both compact and algebraic then its topology is compact and its compact open sets are precisely the open complexes. Moreover, the open complexes form a base for the topology.

Topological fields of sets that are separative, compact and algebraic are called '''Stone fields''' and provide a generalization of the Stone representation of Boolean algebras. Given an interior algebra we can form the Stone representation of its underlying Boolean algebra and then extend this to a topological field of sets by taking the topology generated by the complexes corresponding to the [[Interior algebra#Open and closed elements|open elements]] of the interior algebra (which form a base for a topology). These complexes are then precisely the open complexes and the construction produces a Stone field representing the interior algebra - the '''Stone representation'''. (The topology of the Stone representation is also known as the '''McKinsey–Tarski Stone topology''' after the mathematicians who first generalized Stone's result for Boolean algebras to interior algebras and should not be confused with the Stone topology of the underlying Boolean algebra of the interior algebra which will be a finer topology).