Functorial Quasi-Uniformities on Frames

Abstract

We present a unified study of functorial quasi-uniformities on frames by means of Weil entourages and frame congruences. In particular, we use the pointfree version of the Fletcher construction, introduced by the authors in a previous paper, to describe all functorial transitive quasiuniformities. (2000): 06D22, 54B30, 54E05, 54E15, 54E55. Mathematics Subject Classifications

On Point-finiteness in Pointfree Topology

“…From Propositions 5.2 and 5.5 it follows that the family A of all point-finite covers of a frame L induces, by the method introduced in [6], a transitive quasi-uniformity PF on CL compatible with L. From the following result it follows that A is an adequate kind of covers [7], which, in particular, implies that PF is functorial (and so we may add it to our table of examples in [7]). In order to prove the result we need to recall the well-known fact that, for any frame homomorphism h : L → M , there exists a (unique) frame homomorphism h :…”

The semicontinuous quasi-uniformity of a frame, revisited

“…While the approach via paircovers is most convenient for calculations (the entourage approach asks for a good knowledge of the construction of binary coproducts of frames), the entourage approach allows to formulate the theory directly on frames, in a way very similar to the spatial setting [4,5,17]. For instance, given a frame L, there exists a (entourage) transitive quasiuniformity E on the sublocale frame S(L) which is compatible with L, that is, L 1 (E) = cL (which means that L 1 (E) is an isomorphic copy of the given frame L inside S(L)) [4,5].…”

“…For instance, given a frame L, there exists a (entourage) transitive quasiuniformity E on the sublocale frame S(L) which is compatible with L, that is, L 1 (E) = cL (which means that L 1 (E) is an isomorphic copy of the given frame L inside S(L)) [4,5]. This is the pointfree analogue of the well-known classical fact that for every topological space (X, T) there exists a transitive quasi-uniformity E on X, compatible with (X, T), that is, which induces as its first topology T E the given topology T.…”

“…The semicontinuous quasi-uniformity USC(L) of L is a nice example of a transitive compatible quasi-uniformity [5,6]. The purpose of this paper is to show how the basic facts about USC(L) can be nicely presented with the help of the ring of arbitrary (not necessarily continuous) real-valued functions made available recently by J. Gutiérrez García, T. Kubiak and J. Picado [12].…”

Constructing the sobrification of an approach space via bicompletion

“…Sobriety is an important property since it is exactly what is needed to prove a duality between spaces and frames [2,3,12]. This duality was developed for a sober topological space X and the frame of opens, i.e.…”