The Wallman compactification provides a natural T1-compactification of a T1-space. Given a T1-space X, its Wallman compactification W(X) is the compact T1-space whose points are the minimal prime filters of its topology, endowed with the topology generated by the sets {F | U belongs to F}, for every open U of X.
Although this construction may be regarded as a T1-analogue of the Stone–Čech compactification, its functorial behaviour is more subtle: not every continuous map between T1-spaces lifts to a unique continuous map between the corresponding Wallman compactifications. This naturally raises the question, posed by Herrlich and studied by Harris, of identifying a natural class of maps for which such liftings exist. This led to the class of WC-maps: continuous maps admitting a closed continuous lifting between the corresponding Wallman compactifications. The problem of giving an intrinsic characterization of this class has remained open, to the best of our knowledge.
In this talk, we show that the Wallman construction fits into the general pattern of Stone-type dualities. The key point is that, in the Wallman setting, the relevant points are minimal prime filters rather than prime filters, as in classical Stone duality. This yields a duality between compact T1-spaces with closed continuous maps and a suitable category of bounded distributive lattices.
We then show that this duality extends to a contravariant adjunction between T1-spaces with WC-maps and bounded distributive lattices with suitable morphisms. This leads to an intrinsic characterization of WC-maps, thereby solving Harris's problem.
This is joint work with Mai Gehrke and Matteo Viale.