On box products of syntopogenous spaces

Abstract

The syntopogenous structure is a common generalization of topology, proximity and quasi-uniformity. Actually, the category Snt of syntopogenous spaces contains the category Top of topological spaces, the category Prox of proximity spaces and the category q-Unif of quasi-uniform spaces as bicoreflective subcategories. The category Snt has product which induces the product in Top and in q-Unif, of course. The aim of this paper is to define and investigate another kind of product which gives the box product in Top and in q-Unif. We are grateful to Professor A. Cs~sz~r for some helpful comments and for saving us from several errors. Our basic references are [4] on box products and the monograph [2] on syntopogenous spaces. For convenience we recall the main definitions. A binary relation -< defined on the subsets of a set X is said to be a topogenous order provided that the following conditions are satisfied: (01) O < ~ , X < X , (02) A < B implies A c B , (03) A c A ' < B ' c B implies A<B, (Q) A<B and A'<B" imply AUA'<BUB" and Af~A'<BAB'. A pair (X, S) consisting of a set X and a nonempty family S of topogenous orders on X is called syntopogenous space provided the following axioms hold: (S1) for 1<, 2<E S there is a z<ES with the property 1< U 2 < c 3 < , ($2) for I < E S there is a z<ES with the property ~ < o 2 < D < 1. If S and S' are syntopogenous structures on the set X then S and S" are equivalent provided that for any < E S there is a < ' E S ' such that < c < ' and for any < ' E S ' there is a < E S such that < ' c < . A syntopogenous space (X, S) is topological if S = {<} and < satisfies the following axiom: