Frame quasi-uniformities

Abstract

This paper presents an entourage-like theory of quasi-uniformities for frames. The theory comprises the theory of uniformities for frames as well as the classical theory of quasi-uniformities for spaces. O. IntroductionAccording to JOHN ISBELL [11], if entourages and uniform covers are each used where most convenient in the study of uniform spaces "the result (is) that Tukey's system of uniform covers is used nine-tenths of the time." In the study of quasi-uniform spaces, as opposed to uniform spaces, the roles of covers and entourages are reversed. Quasi-uniform spaces are first defined in terms of entourages by L. NACHBIN [14] and although a cover-like approach for quasiuniform spaces is given by T. GANTNER and R. STEINLAGE [10], the use of Gantner and Steinlage's conjugate-pair covers is even rarer than the one-to-nine ratio Isbell has allotted to entourages in the uniform space setting. A quasi-uniformity on a set X is defined by taking all the axioms for an entourage uniformity except the symmetry axiom. Associated with each quasi-uniform space (X, q/) there are two other quasi-uniformities, the conjugate quasi-uniformity and the join uniformity q/* = q/v @. These quasi-uniformities give rise to three topologies, J(~'), J-(~) and ~-(q/*). In order to provide an appropriate theory of quasi-uniformities for frames, we are 1991 Mathematics Subject Classification: 54E15, 6D20, 18B35.

A pointfree theory of Pervin spaces

“…A (quasi-)uniform homomorphism h : (L, E) → (M, F) is a frame homomorphism h : L → M such that for every E ∈ E we have (h ⊕ h)(E) ∈ F. We denote by QUniFrm the category of quasi-uniform frames and quasi-uniform homomorphisms, and by UniFrm the category of uniform frames and uniform frame morphisms. The following proposition is proven in [3], in which the authors use an alternative definition of quasi-uniform frame. This definition is shown in [14] to be equivalent to the one we use.…”

Untitled

“…In [12,Theorem 3.4] the authors prove that each quasi-uniform frame has a unique completion. In the language of entourage quasi-uniformities [8], if h : (CL, CV) → (L, V) is the completion of (L, V) then {v h * : v ∈ V} is a base for CV. Let v ∈ V and let F be the Weil entourage corresponding to v in the equivalent Weil quasi-uniform structure on L. Then (h ⊕ h) * (F ) corresponds to v h * .…”

“…Examples are the covering uniformities of Isbell [15], the entourage uniformities of Fletcher and Hunsaker [7] and the Weil uniformities of Picado [20]. There are also three equivalent approaches to quasi-uniform frames: Frith [11], Fletcher, Hunsaker and Lindgren [8], and Picado [22]. The equivalence of the approaches to uniform (resp.…”

The Samuel compactification for quasi-uniform biframes

“…On the other hand, in the point-free setting (frames or locales, biframes), the theory of quasi-uniformities was broached using the paircover approach [13,15]; later [11,9] introduced an entourage-like approach and finally the so-called Weil uniformities of [19,26,27,29] provided the direct analogue of entourages.…”