Quasi proximal continuity

Abstract

Conditions are given, under which a quasi-proximally continuous function is quasi-uniformly continuous, or a continuous function is quasi-proximally continuous. Thus, basic results on uniform and proximal continuity are extended and some new results are obtained. Three results in the literature are shown to be false.According to [SI and [9], a quaei-proximity space is a pair (X, 6) , where X is a non empty set and 6 is a binary relation on the power set of X which satisfies: The pair (X, 6) becomes a proximity space when:Pervln showed in LSI that if the closure operator c is defined by c(A) = \x : {x}6A) , then each quasi-proximity space (X, 6) gives rise to a topology r(6) on X and that every topological space {X, r) is quasi-proximizable, that i s , there exists a quasi-proximity 6 on A T such that r(6) = r .Given a quasi-uniformity Q on X (for definitions see [6], [7] or [3]) the quasi-proximity 6 induced by Q is defined by: A6B iff