TY - JOUR
AU - Kelly, J. C.
AB - By J. C. KELLY [Received 15 February 1962] 1. Introduction I T is well known that, when the classical conditions on a metric d( , ) for a set X are relaxed by omitting the requirement d(x, y) = 0 only if z = y, there is no difficulty in generalizing the standard theorems of metric spaces, in particular those concerning metrization of topological spaces. On the other hand, if one attempts to omit the requirement of symmetry, the appropriate generalizations are not obvious. Such unsymmetric distance functions have been studied before by Wilson (9), who used the term quasi-metrics, Ribeiro (6), and others. To my knowledge, there has been no previous attempt to systematize the study of these quasi-metrics. Any distance function d( , ) on a set X (i.e. any non-negative real-valued function d( , ) denned on the product X x X and satisfying the triangle inequality) has the property that the open d-spheres (i.e. sets of the form {y : d(x,y)<t}) form a base for a topology 3) for X. We say that Sd is the topology determined by d( , ). The purpose of this note is to observe that, given on a set 
TI - Bitopological Spaces
JF - Proceedings of the London Mathematical Society
DO - 10.1112/plms/s3-13.1.71
DA - 1963-01-01
UR - https://www.deepdyve.com/lp/wiley/bitopological-spaces-RjCB4z0uaz
SP - 71
EP - 89
VL - s3-13
IS - 1
DP - DeepDyve
ER -