Abstract
The author continues his previous works on preparation to develop generalized axiomatics of the probability theory. The approach is based on the study of set systems of a more general form than the traditional set algebras and their Boolean versions. They are referred to as Dynkin algebras. The author introduces the spectrum of a separable Dynkin algebra and an appropriate Grothendieck topology on this spectrum. Separable Dynkin algebras constitute a natural class of abstract Dynkin algebras, previously distinguished by the author. For these algebras, one can define partial Boolean operations with appropriate properties. The previous work found a structural result: each separable Dynkin algebra is the union of its maximal Boolean subalgebras. In the present note, leaning upon this result, the spectrum of a separable Dynkin algebra is defined and an appropriate Grothendieck topology on this spectrum is introduced. The corresponding constructions somewhat resemble the constructions of a simple spectrum of a commutative ring and the Zariski topology on it. This analogy is not complete: the Zariski topology makes the spectrum of a commutative ring an ordinary topological space, while the Grothendieck topology, which, generally speaking, is not a topology in the usual sense, turns the spectrum of a Dynkin algebra into a more abstract object (site or situs, according to Grothendieck). This suffices for the purposes of the work.
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