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Some Properties of Quasi‐Uniform Spaces

Some Properties of Quasi‐Uniform Spaces By R. STOLTENBERG [Received 17 September 1965] 1. Introduction In a recent publication Kelly (3) studied some of the properties of quasi- metric spaces. 1.1. DEFINITION. A quasi-pseudo-metric for a set X is a non-negative real-valued function d defined on X x X such that, for x, y, z in X, (a) d(x,x) = 0, (b) d(x,y) ^d{x,z) + d(z,y). d is a quasi-metric if, also, (c) d(x, y) = 0 if and only if x = y. Let d be a quasi-(pseudo-)metric on a set, and define d': X x X -> R by the equation d'(x,y) = d{y,x). d' is a quasi-(pseudo-)metric on X. d' and d are called conjugate quasi-(pseudo-)metrics. In this paper the pair (X, d) will be called a quasi-(pseudo-)metric space, and the triple (X, d, d') will be called a bi-quasi-(pseudo-)metric space. The quasi-(pseudo-)metric topology is the family of all sets T such that for each x in T there exists e > 0 with 8 {x) = {y: d(x, y) < e} c T. This topology will be denoted Kelly ((3) 75) defined a sequence {x : n e N} to be d-Cauchy if and only if for every e > 0 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Proceedings of the London Mathematical Society Wiley

Some Properties of Quasi‐Uniform Spaces

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6115
eISSN
1460-244X
DOI
10.1112/plms/s3-17.2.226
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Abstract

By R. STOLTENBERG [Received 17 September 1965] 1. Introduction In a recent publication Kelly (3) studied some of the properties of quasi- metric spaces. 1.1. DEFINITION. A quasi-pseudo-metric for a set X is a non-negative real-valued function d defined on X x X such that, for x, y, z in X, (a) d(x,x) = 0, (b) d(x,y) ^d{x,z) + d(z,y). d is a quasi-metric if, also, (c) d(x, y) = 0 if and only if x = y. Let d be a quasi-(pseudo-)metric on a set, and define d': X x X -> R by the equation d'(x,y) = d{y,x). d' is a quasi-(pseudo-)metric on X. d' and d are called conjugate quasi-(pseudo-)metrics. In this paper the pair (X, d) will be called a quasi-(pseudo-)metric space, and the triple (X, d, d') will be called a bi-quasi-(pseudo-)metric space. The quasi-(pseudo-)metric topology is the family of all sets T such that for each x in T there exists e > 0 with 8 {x) = {y: d(x, y) < e} c T. This topology will be denoted Kelly ((3) 75) defined a sequence {x : n e N} to be d-Cauchy if and only if for every e > 0

Journal

Proceedings of the London Mathematical Society – Wiley

Published: Apr 1, 1967

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