Bitopological Spaces

Abstract

IT 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 X any such (unsymmetric) quasi-metric p( , ) whatsoever, then associated with p(x,y) is the function q{x,y) =p (y,x), and that if one studies the two topologies determined for X by p{ , ) and q( , ), one regains in such 'bitopological spaces' some of the symmetry of the classical metric situation, and in consequence one can obtain systematic generalizations of standard results such as Urysohn's lemma, Urysohn's metrization theorem, Tietze's extension theorem, and the Baire category theorem. The results show in general that the existence of quasi-metrics is related to the existence of real-valued functions which are semicontinuous relative to two topologies in much the same way as the existence of metrics is related to the existence of real-valued functions which are continuous relative to the original topology of the space in question. (The notion of a bitopological space used in relation to semi-continuous functions restores sufficient symmetry to enable one to use some of the existing techniques of continuous functions.)

Zabrejko's lemma and the fundamental principles of functional analysis in the asymmetric case

“…A bitopological space, denoted by (T, τ, ν), is simply a set T endowed with two topologies τ, ν. Bitopological spaces were introduced by Kelly [9] and their study involves some notions relating the topologies τ, ν. A quasi-metric space (X, ρ) can be viewed as a bitopological space with respect to the topologies τ ρ and τρ.…”

Projective bitopological spaces

“…We shall now show that D n {^" 1 ( t / i ) x /~1 ( c / 2 ) } = 0-Suppose these two sets are not disjoint, then let (a 2 …”

SOME PROPERTIES OF $\mathbf{\tau_1\tau_2-\delta}$ SEMI OPEN SETS/CLOSED SETS IN BITOPOLOGICAL SPACES

“…In 1963, J.C. Kelly in [1] initiated the study of bitopological spaces as a natural structure by studying quasimetrics and its conjugate. Besides, he introduced various separation properties into bitopological spaces, and obtained general-izations of some important classical results.…”