We characterize nearness frames whose completions are fine (we call them quotient-fine), and show that the subcategory QfNFrm they form is reflective in the category of strong nearness frames. The resulting functor commutes with the completion functor. QfNFrm is isomorphic to the subcategory of the functor category (RegFrm) 2 given by the dense onto h : M → L, where 2 denotes the category with only two objects and exactly one morphism between them.

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We consider remote points in general extensions of frames, with an emphasis on perfect extensions. For a strict extension ?XL ? L determined by a set X of filters in L, we show that if there is an ultrafilter in X then the extension has a remote point. In particular, if a completely regular frame L has a maximal completely regular filter which is an ultrafilter, then ?L ? L has a remote point, where ?L is the Stone-C?ch compactification of L. We prove that in certain extensions associated with radical ideals and l-ideals of reduced f-rings, remote points induced by algebraic data are exactly non-essential prime ideals or non-essential irreducible l-ideals. Concerning coproducts, we show that if M1 ? L1 and M2 ? L2 are extensions of T1-frames, then each of these extensions has a remote point if the extensionM1?M2?L1?L2 has a remote point.

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We show that the category of almost uniform nearness frames is coreflective in the category of all nearness frames with interpolative uniformly below relation. This we do by actually constructing the coreflection. The resulting functor commutes with the uniform coreflection functor. It also commutes with the totally bounded coreflection functor on the subcategory consisting of nearness frames with strong totally bounded coreflection, but not in general, as a counterexample attests. (2010): Primary 06D22; Secondary 54E15, 54E17. Mathematics Subject Classification