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عنوان البحث(Papers / Research Title)

## Using ?-Operator to Formulate a New Definition of Local Function

الناشر \ المحرر \ الكاتب (Author / Editor / Publisher)

لؤي عبد الهاني جبار السويدي

Citation Information

لؤي,عبد,الهاني,جبار,السويدي ,Using ?-Operator to Formulate a New Definition of Local Function , Time 14/12/2016 09:56:06 : كلية التربية للعلوم الصرفة

وصف الابستركت (Abstract)

In this paper, we use ?-operator in order to get a new version of local function. The concepts of maps, dense, resolvable and Housdorff have been investigated in this paper, as well as modified to be useful in general

الوصف الكامل (Full Abstract)

requirements for our work, we define here the following concepts sequentially: Ideal space, local function, Kuratowski closure, dense, T*-dense, I-dense, codense, ?-operator, resolvable, I-open, pre-I-open, scattered set and Housdorff space. We start define the ideal space. Let (X, T) be a topological space with no separation properties assumed. The topic of ideal topological space has been considered by (Kuratowski, 1966) and (Vaidyanathaswamy, 1960). An ideal I on a topological space (X, T) is a nonempty collection of subsets of X which satisfies the following two condition:
(1) If A I and B A, then B I (heredity).
(2) If A I and B I, then A B I (finite additivity).
Moreover a ?-ideal on (X, T) is an ideal which settle (1), (2) and the following condition:
3.If {Ai : i = 1,2,3,….} I, then {Ai : i =1,2,3,….} I (countable additivity).
An ideal space is a topological space (X, T) with an ideal I on X and is denoted by (X, T, I).For a subset A X, A*( I)={x X : U A I for every U T (x)} is called the local function of A with respect to I and T (Kuratowski, 1933).We simply write A* instead of A*( I) in case there is no chance for disorder. It is familiar that Cl*(A) = A A* defines a Kuratowski closure operator for a topology T *(I) which is finer than T. During this paper, for a subset A X, Cl (A) and In (A) indicate the closure and the interior of A ,respectively. A subset A of an ideal space (X, T, I) is said to be dense (resp, T *-dense (Dontcher, Gansster and Rose, 1999), I –dense (Dontcher, Gansster and Rose,1999) if Cl (A) = X (resp Cl*(A) = X, A*= X).An ideal I on a space (X, T) is said to be codense (Devid, Sivaraj and Chelvam, 2005) if and only if T I = {?}.For an ideal space (X, T, I) and for any A X, where I is codense. Then: dense, T*-dense and I-dense are comparable (Jankovic and Hamlett, 1990). (Natkanies, 1986) used the idea of ideals to define another operator known as ?-operator elucidates as follow: For a subset A X, ? (A) = X-(X-A)*.Equivalently ?(A) = {M T: M-A I}.It is obvious that ?(A) for any A is a member of T. For an ideal space (X, T, I) and Y X then (Y, T y, I y) is an ideal space where T y = {U Y: U T} and I y = {U Y: U I} = {U I: U Y}. In 1943, Hewitt put forward the opinion of a resolvable space as follows: A nonempty topological space (X, T) is said to be resolvable (Hewitt, 1943) if X is the disjoint union of two dense subsets. Given a space (X, T) and A X, A is called I -open (Jankovic and Hamlett, 1990 )(resp per- I –open(Donkhev,1996)if A In (A*)(resp A In Cl*(A).A set A X is called scattered (Jankovic and Hamlett, 1990) i

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