Uniformities and covering properties for partial frames (I)

Partial frames provide a rich context in which to do pointfree structured and unstructured topology. A small collection of axioms of an elementary nature allows one to do much traditional pointfree topology, both on the level of frames or locales, and that of uniform or metric frames. These axioms are sufficiently general to include as examples bounded distributive lattices, $sigma$-frames, $kappa$-frames and frames.  Reflective subcategories of uniform and nearness spaces and lately coreflective subcategories of uniform and nearness frames have been a topic of considerable interest. In cite{jfas9} an easily implementable criterion for establishing certain coreflections in nearness frames was presented. Although the primary application in that paper was in the setting of nearness frames, it was observed there that similar techniques apply in many categories; we establish here, in this more general setting of structured partial frames, a technique that unifies these. We make use of the notion of a partial frame, which is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. After presenting our axiomatization of partial frames, which we call $sels$-frames, we add structure, in the form of $sels$-covers and nearness, and provide the promised method of constructing certain coreflections. We illustrate the method with the examples of uniform, strong and totally bounded nearness $sels$-frames.  In Part (II) of this paper, we consider regularity, normality and compactness for partial frames.