A New Complete Irresoluteness Function AYNUR KESKIN KAYMAKCI Selcuk University, Faculty of Sciences Department of Mathematics 42030, Campus, Konya TURKEY akeskin@selcuk.edu.tr Abstract: In this presentation, first of all we definite a new type of function by using delta-b-open sets. Then, we obtain some characterizations and some properties of this function. Besides, we give their relationships with other types of functions between topological spaces. Key–Words: -b-open sets, b-open sets, -semi-open sets, semi open sets, completely -b-irresolute functions 1 Introduction Of course, the notions of continuous functions and a type of it’s is called irresolute functions are im- portant subject in general topology. So, one can find several papers in literature related it’s. On the other hand, open sets and it’s modifica- tions are studied very authors. One of these sets is b-open. It is defined by Andrijevi´ c [11] and El-Atik [10] independent of each other. It is well-known that b-open set is weaker than semi-open set which is a type of open set. Throughout this paper, we will denote topologi- cal spaces by (X;  ) and (Y;’). For a subset A of a space (X;  ), the closure of A and the interior of A are denoted by Cl(A) and Int(A), respectively. A subset A is said to be regular open ( resp. regu- lar closed ) if Int(Cl(A)) = A ( resp. Cl(Int(A)) = A ). The family of all regular open and regular closed sets of (X;  ) are denoted by RO (X;  ) and RC (X;  ), respectively. A subset A is said to be -open if for each x 2 A there exists a regular open set U such that x 2 U  A. A point x 2 X is called a -cluster point of A if A \ Int(Cl(V )) 6= ? for each open set V containing x. The set of all -cluster points of A is called the -closure of A and is denoted by Cl  (A). The set fx 2 X j x 2 U  A for some regular open set U of X g is called the -interior of A and is denoted by Int  (A). A subset A of a space (X;  ) is called preopen [16] ( resp. b-open [11] or  -open [10], -b-open set [8] ) if A  Int(Cl(A)) ( resp. A  Cl(Int(A)) [ Int(Cl(A)), A  Cl(Int  (A)) [ Int(Cl(A)) ). It is well known that a subset A of a space (X;  ) is called semi open ( resp. -semi-open ) if A  Cl(Int(A)) ( resp. A  Cl(Int  (A)) ). Besides, the complement of -b-open set is said -b-closed. The family of all -b-open and -b-closed sets of (X;  ) are denoted by BO (X;  ) and BC (X;  ), respectively. 2 Completely  -b-irresolute Func- tions In this section, we introduce the notion of completely -b-irresolute functions and obtain some properties of them. Definition 1 A function f : (X;  ) ! (Y;’) is said to be completely -b-irresolute function if the inverse image of each -b-open set V in (Y;’) is regular open set in (X;  ). Now, we give a characterization for completely -b-irresolute functions. Theorem 2 Let f :(X;  ) ! (Y;’) be a function. f is completely -b-irresolute function if and only if the inverse image of each -b-closed set F in (Y;’) is regular closed in (X;  ). Proof. The proof is obvious by considering the com- plement of Definition 1. Definition 3 A function f :(X;  ) ! (Y;’) is said to be completely irresolute [1] ( resp. com- pletely -semi-irresolute [2], completely b-irresolute ) if f 1 (V ) is regular open set in (X;  ) for every semi open ( resp. -semi-open, b-open ) set V in (Y;’). Remark 4 For a function f :(X;  ) ! (Y;’), we have the following diagram by using Definitions 1 and 2. WSEAS TRANSACTIONS on MATHEMATICS Aynur Keskin Kaymakci E-ISSN: 2224-2880 523 Volume 15, 2016