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Uniformities and covering properties for partial frames (I) مقاله



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 suciently general to include as examples bounded distributive lattices, -frames, -frames and frames. Re ective subcategories of uniform and nearness spaces and lately core- ective subcategories of uniform and nearness frames have been a topic of considerable interest. In [9] an easily implementable criterion for establishing certain core ections 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 unies 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 nite meets distribute over these. After presenting our axiomatization of partial frames, which we call S-frames, we add structure, in the form of S-covers and nearness, and provide the promised method of constructing certain core- ections. We illustrate the method with the examples of uniform, strong and totally bounded nearness S-frames. In Part (II) of this paper ([10]) we consider regularity, normality and compactness for partial frames.

کلیدواژه ها:

frame ، Near ، Normal ، ness ، Strong ، Uniformity ، partial frame ، meet-semilattice ، strong inclusion ، uniform map ، core ection ، totally bounded ، regular ، compact. Mathematics ، S-frame ، Z-frame ، P-approximation

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