Strongly proximal continuity & strong connectedness

This article introduces strongly proximal continuous (s.p.c.) functions, strong proximal equivalence (s.p.e.) and strong connectedness. A main result is that if topological spaces X, Y are endowed with compatible strong proximities and f : X -> Y is a bijective s.p.e., then its extension on the hyperspaces CL(X) and CL(Y), endowed with the related strongly hit and miss hypertopologies, is a homeomorphism. For a topological space endowed with a strongly near proximity, strongly proximal connectedness implies connectedness but not conversely. Conditions required for strongly proximal connectedness are given. Applications of s.p.c. and strongly proximal connectedness are given in terms of strongly proximal descriptive proximity. (C) 2016 Elsevier B.V. All rights reserved.