 Remarkably, the properties listed are exactly the fundamental axioms of an important branch of mathematics known as  point-set  topology [Abramsky 1987, Vickers 1986]. A topological space is defined by a collection of sets known as  open sets or neighbourhoods, that satisfy the axiomatic properties of verifiable hypotheses (closure under arbitrary union and finite disjunction, both the empty set and the entire space are open). The set-theoretic complements of open sets are called closed sets, so refutable hypotheses correspond exactly to closed sets. Point-set topology was invented to support a kind of generalized functional analysis without numbers (more precisely, without distances). It is striking that the foundational axioms of topology have an exact epistemological interpretation in terms of the properties of empirical hypotheses that allow verification or falsification with certainty. Current mathematical developments of learning theory often begin by taking as a basic concept a set of verifiable hypotheses satisfying the properties listed. This approach has two advantages.