Strongly Ɵcl-continuous functions

Author
Masoumeh Etebar
Shahid Chamran University of Ahvaz
Abstract
. 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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Mathematical Research
Volume 9, Issue 2 - Serial Number 25
Summer 2023
Pages 107-124