I am making two papers available as Departmental Research Reports: "Topical Categories of Domains" "Localic Completion of Quasimetric Spaces" Both explore the idea of locales (and, indeed, toposes) as "topology-free spaces". The technique is to work not with frames (point-free topologies) but with presentations of them, understood as propositional geometric theories whose models are the points. (But it is normally more convenient to work with equivalent predicate theories.) Then - * the geometric theory already determines an implicit topology on its models; * any construction of models of one theory out of models of another automatically determines a continuous map (or geometric morphism), just so long as the construction is geometric. In effect, a restriction to geometric mathematics removes the need to treat topology explicitly, hence "topology-free spaces". Apparently, explicit topology is needed to correct the over-credulousness of classical reasoning principles, though in practice the geometric constraints often end up forcing one to reintroduce the normal topological arguments in a different guise. The two papers test the applicability of the idea in the two areas of domain theory and quasimetric spaces. Aside from the "topology-free space" aspects, the papers develop some new results: "Topical Categories of Domains" addresses categorical domain theory and replaces the usual classes of objects and morphisms by toposes classifying them. New general results concerning fixpoints of endo-geometric-morphisms of toposes exploit their intrinsically topological nature to give a simple approach to the solution of domain equations. The paper also gives a summary of the constructive theory of Kuratowski finite sets and establishes some limitations to the Cartesian closedness of Sets. "Localic Completion of Quasimetric Spaces" proposes a construction of locales in completion of quasimetric spaces (using ideas of flatness deriving from Lawvere's enriched category account), studies the powerlocales and shows that a limit map from a locale of Cauchy sequences to the completion is triquotient in the sense of Plewe. Paper copies are available from me; electronic copies are expected to be available shortly in the Department of Computing's Research Report series coordinated by Frank Kriwaczek (frk@doc.ic.ac.uk). Steve Vickers.