Uniformities and covering properties for partial frames (I)

Abstract. 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.
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.
Frame, S-frame, Z-frame, partial frame, -frame, -frame, meet-semilattice, near- ness, uniformity, strong inclusion, uniform map, core ection, P-approximation, strong, totally bounded, regular, normal, compact. Mathematics Subject Classication [2010]: 0



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