## 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, Lindelöf, etc.)

### Citations:

Zbl 1010.54012; Zbl 1081.54011; Zbl 0542.54011; Zbl 0435.54010
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