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R-supercontinuous functions

Kohli J. K., Singh D., Aggarwal J.

Demonstratio Mathematica

2010

Vol. 43, nr 3

703-723

A new strong variant of continuity called 'R-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 R-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 (...)-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.