Entourage uniformities for frames

Fletcher, Peter; Hunsaker, W.

doi:10.1007/bf01351768

Cited by 16 publications

(10 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A quasi-uniformity U is called proximally symmetric or Smyth symmetric [7] provided that its induced quasiproximity is a proximity. Of course, a proximally symmetric totally bounded quasi-uniformity is a uniformity.…”

Section: Infima Of Quasi-uniformitiesmentioning

confidence: 99%

Infima and complements in the lattice of quasi-uniformities

Jager

Künzi

2007

Topology and its Applications

View full text Add to dashboard Cite

We show that the infimum of any family of proximally symmetric quasi-uniformities is proximally symmetric, while the supremum of two proximally symmetric quasi-uniformities need not be proximally symmetric. On the other hand, the supremum of any family of transitive quasi-uniformities is transitive, while there are transitive quasi-uniformities whose infimum with their conjugate quasi-uniformity is not transitive. Moreover we present two examples that show that neither the supremum topology nor the infimum topology of two transitive topologies need be transitive. Finally, we prove that several operations one can perform on and between quasi-uniformities preserve the property of having a complement.

show abstract

Section: Infima Of Quasi-uniformitiesmentioning

confidence: 99%

Infima and complements in the lattice of quasi-uniformities

Jager

Künzi

2007

Topology and its Applications

View full text Add to dashboard Cite

show abstract

“…In [16] Isbell introduced uniformities on frames, as the precise translation into frame terms of Tukey's notion, later developed in detail by Pultr [25]. We note that, as in the case of spaces, there are several different ways of describing uniformities on frames, such as the functional uniformities of Fletcher-Hunsaker ( [7], [8]) and the entourage unifromities of Picado [21].…”

Section: Uniform Framesmentioning

confidence: 99%

Uniform-type structures on lattice-valued spaces and frames

García

Mardones-Pérez

Picado

et al. 2008

Fuzzy Sets and Systems

View full text Add to dashboard Cite

By introducing lattice-valued covers of a set, we present a general framework for uniform structures on very general L-valued spaces (for L an integral commutative quantale). By showing, via an intermediate L-valued structure of uniformity, how filters of covers may describe the uniform operators of Hutton, we prove that, when restricted to Girard quantales, this general framework captures a significant class of Hutton's uniform spaces. The categories of L-valued uniform spaces and L-valued uniform frames here introduced provide (in the case L is a complete chain) the missing vertices in the commutative cube formed by the classical categories of topological and uniform spaces and their corresponding pointfree counterparts (forming the base of the cube) and the corresponding L-valued categories (forming the top of the cube).

show abstract

“…A frame is uniformizable provided that it admits an entourage uniformity as defined in [4]. Using the axiom of countable dependent choice, A. PULTR has shown that a flame is uniformizable if and only if it is completely regular [15,Th.…”

Section: Preliminariesmentioning

confidence: 99%

“…Definition [4]. Let L be a frame and let B be a base for a frame quasi-uniformity on L. If B satisfies:…”

Section: Preliminariesmentioning

confidence: 99%

“…Since the U-V quasi-uniformly continuous maps are U v U-V v ~" uniformly continuous and since every frame uniformity is a quasi-uniformity, our theory comprises the theory of uniformities given in [4]. We note that if (X, q/) is a quasi-uniform space we can define a quasi-uniformity on the frame ~(X) or on the frame r v ~) and in either case recover the classical theory.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Frame quasi-uniformities

Fletcher

Hunsaker

Lindgren

1994

Monatshefte f�r Mathematik

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Entourage uniformities for frames

Cited by 16 publications

References 6 publications

Infima and complements in the lattice of quasi-uniformities

Infima and complements in the lattice of quasi-uniformities

Uniform-type structures on lattice-valued spaces and frames

Frame quasi-uniformities

Contact Info

Product

Resources

About