ON d-I-proper function

Abstract

In this paper,we introduce a new class of function called δ-I-proper function and explain some of propositions,theorems and some equivalent statements of this function . Key words and phrases: -I-cluster point,R-I-open set ,-I-open set , -I-continuous function,-I-closed function,-I-proper function . 1- Introduction: Let A be a subset of a topological space (X,T).The δ-interior of a subset A of X is the union of all regular open (R-open) sets of contained in A and is denoted by Int(A) [7] . The subset A is called δ-open if A=Int(A) (i.e. a set A is δ-open if it is the union of regular open sets )[7] .The complement of a δ-open set is called δ-closed . Alternatively , a set A(X,T) is called δ-closed if A=Cl(A), where Cl(A)={xX:Int(Cl(U))⋂A, UT and xU}[1].The family of all δ-open sets forms a topology on X and denoted by T .Since the intersection of two regular open sets is regular open , the collection of all regular open sets forms a base for a coarser topology TS than the original one T . A basic facts are that TS= T and every clopen set is regular open set [7] . In a topological space (X,T) , let I an ideal of subsets of X .An ideal is defined as a non-empty collection I of subsets of X satisfying two conditions : 1) If AI and BA , then BI ; 2) If AI and BI , then ABI .A topological (X,T) with an ideal I on X is called ideal topological space and denoted by (X,T,I) . For a subset AX , A={xX:U⋂AI , UT and xU} is called the local function of A with respect to I and T [3] .For each ideal topological space (X,T,I) , there exists a topology T finer than T , generated by (I,T)={U\I:UT and II}, but in general is not always a topology [2].Additionally , Cl(A)=AA defines a Kuratowiski closure for T and T T .