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.


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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