R-supercontinuous functions.

*(English)*Zbl 1217.54016Summary: 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, LindelĂ¶f, etc.) |