The (positive) unique structure theorem

There is an amazing theorem in the theory of analytic spaces, called the unique structure theorem: any two countable families F₁ and F₂ of Borel subsets of an analytic space X that separate the points of X generate the same σ-algebra. F. Clerc, N. Fijalkow, B. Klin and P. Panangaden used a generalization of this in a 2019 paper about labelled Markov processes, which they call the positive unique structure theorem. It too, applies to any analytic space, and I will show that it even generalizes to a larger class of spaces, which I will call quasi-analytic; those are the continuous images of M. de Brecht’s quasi-Polish spaces (inside an arbitrary topological space, not necessarily Hausdorff or even T₀). The treatment I will give is a mix of measure-theoretic and domain-theoretic techniques. Read the full post.