On the representation of uniform structures by extended reticles

An extended reticle R on a topological space X is a nonempty family of ordered pairs (A,B) such that (i) A is closed and B is open, (ii) A⊂B, (iii) if (A,B)∈R then (X−B,X−A)∈R, (iv) (A,B)∈R implies the existence of an open set B∗ and a closed set A∗ such that (A,B∗)∈R, B∗⊂A∗ and (A∗,B)∈R, (v) {(Ai,Bi):i∈I} is in R and {Bi:i∈I} is locally finite implies (⋃i∈IAi,⋃i∈IBi)∈R. If in (v) |I|≤k. then it is called a k-extended reticle. Main results: (1) Every proximity class of uniformities contains at most one k-uniformity associated with a k-extended reticle; if it contains one then it is the finest member. (2) X is pseudocompact if and only if the finest precompact uniformity on X having topology coarser than that of X is the א0-uniformity associated with an א0-extended reticle on X. (3) There is a proximity class of uniformities on the real line (with the usual topology) such that none of its members is the k-uniformity associated with a k-extended reticle, k an arbitrary infinite cardinal. This last result answers negatively a question posed by G. Aquaro [Ann. Mat. Pura Appl. (4) 47 (1959), 319–389