In this paper, we introduce a new class of open sets in a topological space called − open sets. We study some properties and several characterizations of this class, also we explain the relation of − open sets with many other classes of open sets. Furthermore, we define − closed sets and − closed sets and we give some fundamental properties and relations between these classes and other classes such as − closed and − closed sets.

Following Caldas in [1] we introduce and study topological properties of ii-derived, ii-border, ii-frontier, and ii-exterior of a set using the concept of ii-open sets. Moreover, we prove some further properties of the well-known notions of ii-closure and ii-interior. We also study a new decomposition of ii-continuous functions. Finally, we introduce and study some of the separation axioms specifically 0ii T , 1ii T .

In this paper, we define ii-open set in bitopological space as follows: Let ( , 1 , 2 ) be a bitopological space, a subset A of is said to be ( 1or A⊆ 2 ( ∩ ) We study some characterizations and properties of this class. Also, we explain the relation between ii-open sets and open sets, i-open sets and α-open sets in bitopological space. Furthermore, we define ii-continuous mapping on bitopological spaces with some properties.

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A new class of separation axioms known as ic-separation axioms is introduced in this study. They rely on new extended open sets known as ic-open sets, and we describe the relationship between them and give several examples. Additionally, we define ic-generalized closed set concepts in a topological space in order to frame the another class of separation axioms called ic-generalized separation axioms. Among other things, the basic concern properties and relative preservation properties of these spaces are projected under ic-generalized irresolute mappings.