A method for constructing coreflections for nearness frames. (Q2254580)

This paper provides a method for constructing coreflections in the category of nearness frames. A key ingredient is the complete lattice of all sub nearness frames of a nearness frame. The notion of a sub nearness frame is introduced in the paper: a nearness frame \((L,\mathcal NL)\) is a \textit{sub nearness frame} of a nearness frame \((M,\mathcal NM)\) if \(L\) is a subframe of \(M\) and \(\mathcal NL\subseteq\mathcal NM\). The method unifies such important cases as the well-known uniform, totally bounded and separable coreflections. The power of the method is further demonstrated by the result that any full, isomorphism-closed coreflective subcategory of the category of nearness frames for which the coreflection maps are all one-to-one, can be obtained by it. As an application, the authors use the method to show that strong nearness frames are coreflective in nearness frames, solving an open problem of \textit{T. Dube} and \textit{M. M. Mugochi} [Quaest. Math. 34, No. 2, 247-263 (2011; Zbl 1274.06039)]. Finally, they explain how the method also applies to some other categories such as the categories of prenearness frames and nearness \(\sigma\)-frames.