|My mathematical work centres on locale theory. In collaboration with Vickers, in 2002, we showed that any Scott continuous map between frames can be represented by a natural tranformation. The main point of this observation is that continuity becomes categorical and so independent of set theory. This independence from set theory allows us to develop a topological theory within which compact Hausdorff is order dual to discrete. We are able to develop a theory that simulataneously (and dually) describes the behaviour of compact Hausdorff spaces and sets.
The following results and definitions are all independent of set theory
(a) the Hofmann-Mislove theorem
The duality can either connect two seemingly different results, showing them to be the same, or it can prove new results. For example the dual of (a) is the Bung/Funk constructive description of the points of the lower power locale and the dual of (b) is a new result for constructive locale theory which is that every compact subgroup of a localic group is fitted. The notion of geometric morphism and triquotient surjection are self-dual.