Mathematical Researches

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. The class of strongly Ɵcl - continuous functions is considered in this paper. Studying about basic properties of strongly Ɵcl - continuous functions, it is observed that these properties are similar to the properties of continuous functions. We denote by Scl(X)  the ring of all real valued strongly Ɵcl - continuous functions on a topological space X . It is proved that if the range of a strongly  Ɵcl- continuous function f  is a T1 - space, then f  is constant on quasi – components of its domain. Using this fact, we prove that for every topological space X , there is an ultra- Hausdorff  space Y  such that Scl(X) is isomorphic to C(Y).The behavior of these functions in relation to separation axioms is studied. We show that if f is a strongly Ɵcl - continuous function from a topological space X  to a T0 - space Y , then X  is  ultra-Hausdorff. Topological properties of direct and inverse image of spaces with certain topological properties under strongly Ɵcl - continuous functions are investigated. Among them, it is proved that the image of every cl-closure compact space under a strongly Ɵcl - continuous function is compact. Finally the properties of the graphs of strongly Ɵcl - continuous functions are discussed. It is proved that for every compact and Hausdorff space Y , strong Ɵcl - continuity of the function f from X to Y is equivalent to Ɵcl - closedness of  the graph of f relative to X .                                                   

Keywords: Ɵcl - open set, Ɵcl -closed set, Ɵcl -space, cl - closure compact space, Ɵcl -closed graph

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