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Rδ-supercontinuous functions

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

Demonstratio Mathematica

2014

Vol. 47, nr 2

433--448

A new class of functions called ‘Rδ-supercontinuous functions’ is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity which already exist in the literature is elaborated. The class of Rδ-supercontinuous functions (Math. Bohem., to appear) properly contains the class of Rz-supercontinuous functions which in its turn properly contains the class of Rcl-supercontinuous functions (Demonstratio Math. 46(1) (2013), 229–244) and so includes all cl-supercontinuous (=clopen continuous) functions (Applied Gen. Topol. 8(2) (2007), 293–300; Indian J. Pure Appl. Math. 14(6) (1983), 767–772) and is properly contained in the class of R-supercontinuous functions (Demonstratio Math. 43(3) (2010), 703–723).

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.