case where every subset is an event.) Measure spaces and probability spaces play a foundational role in measure theory and probability theory respectively. In applications to Physics we often deal with measure spaces and probability spaces derived from rich mathematical structures such as inner product spaces or topological groups which already have a topology associated with them - this should not be confused with the topology generated by taking arbitrary unions of complexes. Topological fields of sets A topological field of sets is a triple where is a topological space and is a field of sets which is closed under the closure operator of or equivalently under the interior operator i.e. the closure and interior of every complex is also a complex. In other words, forms a subalgebra of the power set interior algebra on Topological fields of sets play a fundamental role in the representation theory of interior algebras and Heyting algebras. These two classes of algebraic structures provide the algebraic semantics for the modal logic S4 (a formal mathematical abstraction of epistemic logic) and intuitionistic logic respectively. Topological fields of sets representing these algebraic structures provide a related topological 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 companions of intermediate logics. Given a topological space the 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