A duality involving Borel spaces
A Borel space is a set together with a collection of subsets (called Borel subsets) which is closed under countable unions and complements. Borel maps are functions between the underlying sets the inverse images of which preserve Borel sets. The arising category is called the category of Borel spaces. Sobriety in this category is defined in a manner fairly akin to one of characterizations of sober topological spaces. The gist of the paper is in establishing a dual equivalence between sober Borel spaces and spatial Boolean \(\sigma\)-frames. Reviewer's remark: It appears that there is an inadvertent tautology in Corollary 3.4. Since the Euclidean space \(\mathbb R^n\) is a separable metric space, and any metric space is separable iff it is second countable, statements (1) and (2) of the corollary are subsumed by statement (3).
scientific article; zbMATH DE number 5318671
scientific article