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**Generalizations of \(Z\)-supercontinuous functions and \(D_\delta\)-supercontinuous functions.**
*(English)*
Zbl 1181.54020

Summary: Two new classes of functions, called ‘almost \(z\)-supercontinuous functions’ and ‘almost \(D_\delta\)-supercontinuous functions’ are introduced. The class of almost \(z\)-supercontinuous functions properly includes the class of \(z\)-supercontinuous functions [J. K. Kohli and R. Kumar, Indian J. Pure Appl. Math. 33, No. 7, 1097–1108 (2002; Zbl 1010.54012)] as well as the class of almost clopen maps [E. Ekici, Acta Math. Hung. 107, No. 3, 193–206 (2005; Zbl 1081.54011)] and is properly contained in the class of almost \(D_\delta\)-supercontinuous functions which in turn constitutes a proper subclass of the class of almost strongly \(\theta\)-continuous functions [T. Noiri and S. M. Kang, Indian J. Pure Appl. Math. 15, 1–8 (1984; Zbl 0542.54011)] and which in its turn include all \(\delta\)-continuous functions [T. Noiri, J. Korean Math. Soc. 16, 161–166 (1980; Zbl 0435.54010)]. Characterizations and basic properties of almost \(z\)-supercontinuous functions and almost \(D_\delta\)-supercontinnous functions are discussed and their place in the hierarchy of variants of continuity is elaborated. Moreover, properties of almost strongly \(\theta\)-continuous functions are investigated and sufficient conditions for almost strongly \(\theta\)-continuous functions to have \(u_\theta\)-closed (\(\theta\)-closed) graph are formulated.

### MSC:

54C05 | Continuous maps |

54C08 | Weak and generalized continuity |

54C10 | Special maps on topological spaces (open, closed, perfect, etc.) |

54D10 | Lower separation axioms (\(T_0\)–\(T_3\), etc.) |

54D15 | Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) |

54D20 | Noncompact covering properties (paracompact, LindelĂ¶f, etc.) |