Uniformities and covering properties for partial frames (II)

This paper is a continuation of [Uniformities and covering properties for partial frames (I)], in which 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 there our axiomatization of partial frames, which we call sels-frames, we added structure, in the form of sels-covers and nearness.  Here, in the unstructured setting, we consider regularity, normality and compactness, expressing all these properties in terms of sels-covers. We see that an sels-frame is normal and regular if and only if the collection of all finite sels-covers forms a basis for an sels-uniformity on it. Various results about strong inclusions culminate in the proposition that every compact, regular sels-frame has a unique compatible sels-uniformity.

Keywords:

frame , sels-frame , Z-frame , partial frame , sigma-frame , kappa-frame , meet-semilattice , nearness , Uniformity , strong inclusion , uniform map , coreflection , P-approximation , strong , totally bounded , regular , normal , compact