A short note on the role of grills in nearness frames

Abstract

Grills and clusters have featured quite significantly in the theory of nearness spaces (see for example H. Herrlich (Topological structures, Topological Structures 1. Math. Centre Tracts, 59–122, 1974) and G. Preuss, (Theory of topological structures: An approach to categorical topology (D. Reidel Pub. Co., 1987)). In this paper we consider the role they play in nearness frames. Some of the results we establish here are that, in a separated Boolean nearness frame, a near grill is contained in a unique cluster, and that, in quotient-fine nearness frames with spatial completion, near subsets are contained in near grills.