Utilizing Operation in Certain Types of Open Sets A Review

Abstract

The idea of operation was initiated for the first time by Kasahara[1], and he investigated the properties of -continuous functions and -closed graph of operation . After then Krishnan and Balachandran[2], [3] applied operation  to define a new type of semi-open and pre-open namely  semi-open and  pre-open sets. Consequently, Hariwan[4] gave an overview of b-open set and  b-continuous functions along with shown some general characteristics of various separation axioms.  In 2001, Maki and Noiri [5] used bi-operation notion to study -open sets which several characterizations are obtained. Since that many researchers have identified various kinds of open sets regarding different topologies by using operation.

Conclusion:

The notion of operation has an essential position to describe new concepts in topology such as continuity, compactness, and separation axioms. In this work, we recalled the definitions of various types of open sets regarding operation and overview the relations among these sets and compare the results through previous studies. Later, many researchers were interested to establish new types of open sets regarding to bi-operation instead of using operation. Comparisons of different open sets regarding operation and bi-operation have been made. Furthermore, we summarized some applications for these sets for operation and bi-operation and explained their relationships. Finally, it would be useful to use this principle to research these features in soft and micro topology in the future.