Generalised Stone Dualities

Point-free topology traces its origins to Stone's classic duality between Boolean algebras and compact Hausdorff 0-dimensional spaces.  To deal with non-0-dimensional spaces, the focus shifted from Boolean algebras, representing a mere basis of the space, to frames, representing the entire open set lattice.  In this talk we outline new ways of showing how bases can in fact still be used to more faithfully extend Stone's classic duality to non-0-dimensional spaces. (joint work with Charles Starling)