Abstract. A new strong variant of continuity called 'i?-supercontinuity' is introduced. Basic properties of R-supercontinuous functions are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. It is shown that fl-supercontinuity is preserved under the restriction, shrinking and expansion of range, composition of functions, products and the passage to graph function. The class of .R-supercontinuous functions properly contains each of the classes of (i) strongly 0-continuous functions introduced by Noiri and also studied by Long and Herrington; (ii) D-supercontinuous functions; and (iii) F-supercontinuous functions; and so include all z-supercontinuous functions and hence all clopen maps (= cl-supercontinuous functions) introduced by Reilly and Vamnamurthy, perfectly continuous functions defined by Noiri and strongly continuous functions due to Levine. Moreover, the notion of r-quotient topology is introduced and its interrelations with the usual quotient topology and other variants of quotient topology in the literature are discussed. Retopologization of the domain of a function satisfying a strong variant of continuity is considered and interrelations among various coarser topologies so obtained are observed.

Three weak variants of compactness which lie strictly between compactness and quasicompactness, are introduced. Their basic properties are studied. The interplay with mapping and their direct and inverse preservation under mappings are investigated. In the process three decompositions of compactness are observed.

Abstract.Reilly and Vamanamurthy introduced the class of 'clopen maps' (≡ 'cl-supercontinuous functions'). Subsequently generalizing clopen maps, Ekici defined and studied almost clopen maps (≡ almost cl-supercontinuous functions). Continuing in the spirit of Ekici, here basic properties of almost clopen maps are studied. Behavior of separation axioms under almost clopen maps is elaborated. The interrelations between direct and inverse transfer of topological properties under almost clopen maps are investigated. The results obtained in the process generalize, improve and strengthen several known results in literature including those of Ekici, Singh, and others.

We extend the notion of R-weak commutativity and its variants to probabilistic metric spaces and prove common xed point theorems concerning them. Examples are included to reect upon the distinctiveness of the types of mappings dened in the paper.