Editing Field of sets (section)

==== 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).