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R-supercontinuous functions. (English) Zbl 1217.54016
Summary: 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 functions. The class of $$R$$-supercontinuous functions properly contains each of the classes of (i) strongly $$\theta$$-continuous functions introduced by Noiri and also studied by Long and Herrington; (ii) $$D$$-supercontinuous functions; and (iii) $$F$$-supercontinuous functions; and so includes all $$z$$-supercontinuous functions and hence all clopen maps ($$\equiv$$ 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.

##### MSC:
 54C08 Weak and generalized continuity 54C10 Special maps on topological spaces (open, closed, perfect, etc.) 54D10 Lower separation axioms ($$T_0$$–$$T_3$$, etc.) 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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