Set operators and associated functions

https://com munications.science.ankara.edu.tr Com mun.Fac.Sci.Univ.Ank.Ser. A1 M ath. Stat. Volum e 70, Numb er 1, Pages 456–467 (2021) DOI: 10.31801/cfsuasm as.644689 ISSN 1303–5991 E-ISSN 2618–6470 Received by the editors: N ovem ber 9, 2019; Accepted: M arch 3, 2021 SET OPERATORS AND ASSOCIATED FUNCTIONS Shyamapada MODAK and Sk SELIM Department of Mathematics, University of Gour Banga P.O. Mokdumpur, Malda 732 103, INDIA Abstract. The study of two operators local function and the set operator on the ideal topological spaces are likely to be the same as the study of closure and interior operator of the topological spaces. However, they are not exactly equal to the interior and closure operator of the topological spaces. In this context, we introduce two new set operators on the ideal topological spaces. Detailed properties of these two operators are the part of this article. Furthermore, the operators interior (resp. ) and closure (local function) obey the relation Int(A) = X n Cl(X n A) (resp. (A) = X n (X n A) ). We search the general method of these relations, through this manuscript. 1. Introduction and Preliminaries Let X be a set and }(X) be the power set of X. A sub-collection I of }(X) is said to be an ideal [9,21] on X if I has hereditary and …nite additivity property. If I is an ideal on a set X and is the topology on the same set X, then the triplicate (X; ; I) is said to be an ideal topological space [3]. We, throughout the paper denoted, it by G. Since G deals with two mathematical structures: ideal I and topology simultaneously, thus the condition \ I = f;g has played important role for the study of the ideal topological space. This condition termed as codense ideal [4] or -boundary [5, 17] or Hayashi-Samuel space [3]. Common representation of various types limit points like condensation point, accumulation point, !-accumulation point of the topological space is the local function [6, 7, 9] of the ideal topological space G. For a subset A of X, the local function is: A = fx 2 X j U \ A 2 = Ig, where U 2 (x) = fx 2 U j U 2 g. For the detail study of the local function, Natkaniec [18] had introduced another set operator which is called operator. This operator is de…ned as: for an ideal topological space G and for A X, (A) = X n (X n A) . This relation is similar to the well know relation of the 2020 Mathematics Subject Classi…cation. Primary 54A05; Secondary 47H04. Keywords and phrases. Local function, operator, associated function, M operator. -Corresponding author; 0000-0002-0226-2392; 0000-0002-4226-2004. c 2 0 2 1 A n ka ra U n ive rsity C o m m u n ic a tio n s Fa c u lty o f S c ie n c e s U n ive rsity o f A n ka ra -S e rie s A 1 M a th e m a tic s a n d S ta tistic s 456 SET OPERATORS AND ASSOCIATED FUNCTIONS 457 topological space (X; ): Int(A) = X n Cl(X n A), where A X, and ‘Int’ and ‘Cl’ denoted as the interior and closure operator respectively of the topological space. Note that the ‘local function’ is not a closure operator and the operator is not an interior operator. However, the operator l(A) = A [ A makes a closure operator [6, 7, 9] and it induces a topology on X. This topology is called -topology [17, 19] on X and it is denoted as (I) [8, 10, 14, 16, 20] (or simply ). The closure operator of the -topology is denoted as Cl and the interior operator of the -topology is denoted as Int . Furthermore, Int (A) = A \ (A) [2,5,14,16]. This present paper has divided into two parts: one part is some new type of set operators on the ideal topological spaces and their relations. Characterizations of the Hayashi-Samuel spaces is also included in this part. The another part of this paper is related to the set-theory. Actually through this part we search the generalization of the following relations: (A) = X n (X n A) , Int(A) = X n Cl(X n A) and A M = X n M (X n A) [13] etc. Before starting main section we need following tools from the literature: Proposition 1. [8] Let G be an ideal topological space. The followings are equivalent. (1) X = X; (2) \ I = f;g; (3) If I 2 I, then int(I) = ;; (4) For every U 2 ; U U . Lemma 2. [14] Let G be a Hayashi-Samuel space. Then for A X, (A) A . Proposition 3. [5, 8] Let G be an ideal topological space, and A and B be two subsets of X. Then the following properties hold: (1) If A B, then A B ; (2) A = Cl(A ) Cl(A), (3) (A ) A , (4) (A [ B) = A [ B , (5) If I 2 I, then (A [ I) = A = (A n I) , (6) If U 2 , then U (U ). De…nition 4. A set-valued set function p : }(X) ! }(X) is said to be grounded (resp. idempotent) if p(;) = ; (resp. p(p(A)) = p(A)), where A 2 }(X). 2. The operator r1 We de…ne the operator r1 on an ideal topological space (X; ; I) in the following way: for a subset A of X, r1 (A) = Y(A) \ ^(A), where Y(A) = (A) n A and ^(A) = (A) n A. For the ideal topological space G, if I = f;g (resp. I = }(X)), then r1 (;) = ; (resp. (A) = X n A for any subset A of X). 458 S. M ODAK, S. SELIM Lemma 5. Let G be an ideal topological space and A be a subset of X. Then r1 (A) = (A) n (A [ A). Proof. r1 (A) = ( (A) n A ) \ ( (A) n A) = [ (A) \ (X n A )] \ [ (A) \ (X n A))] = (A) \ [(X n A ) \ (X n A)] = (A) \ [X n (A [ A)] = (A) n (A [ A). Theorem 6. Let G be an ideal topological space. Then following statements hold: (1) for all A; B X, (A [ B) r1 (A) \ r1 (B). (2) for all A X, r1 (A) = (A) n (Cl(A ) [ A). (3) for all A X, r1 (A) (A) n (Cl(A) [ A). (4) for all A X, r1 (A) = (A) n Cl (A). (5) for all U 2 , r1 (U ) = ;. (6) for all U 2 , r1 (U ) = ;. (7) for all A X, r1 (X n A) = Y(A) \ Z(A), where Z(A) = (A n A ). (8) for all A X, (r1 (A)) = A [ X . (9) for all A X, r1 (A) 2 . Proof. 1. We know (A) n A (A [ B) n A and (A) n A (A [ B) n A then [ (A) n A ] \ [ (A) n A] [ (A [ B) n A ] \ [ (A [ B) n A]. Therefore r1 (A) (A[B). Similarly r1 (B) (A[B). Hence r1 (A)\r1 (B) (A[B) i.e., (A [ B) r1 (A) \ r1 (B). 2. From Lemma 5, r1 (A) = (A) n (A [ A) and from Proposition 3, A = Cl(A ). Therefore r1 (A) = (A) n (Cl(A ) [ A). 3. From Lemma 5, r1 (A) = (A) n (A [ A) and from Proposition 3, Cl(A ) Cl(A). Therefore r1 (A) (A) n (Cl(A) [ A). 4. From Lemma 2, r1 (A) = (A) n (A [ A). This implies that r1 (A) = (A) n Cl (A). 5. We have r1 (U ) = ( (U ) n U ) \ ( (U ) n U ) and ^(U ) = (U ) n U . Hence for U 2 , r1 (U ) (U ) n U = ;, U (U ) [5]. 7. r1 (X n A) = [ (X n A) n (X n A) ] \ [ (X n A) n (X n A)] = [(X n A ) n (X n A) ] \ [(X n A ) n (X n A)] = ( (A) n A ) \ (A n A ) = Y(A) \ Z(A). 8. We have, (r1 (A)) = X n (X n r1 (A)) . Now X n [ (A) n Cl (A)] = X n [X n (X n A) n Cl (A)] = X n [X n Cl (A) [ (X n A) ] = Cl (A) [ (X n A) = A [ A [ (X n A) = A [ [A [ (X n A)] = A [ X . Corollary 7. Let G be an ideal topological space. Then for any A (r1 (A)). X, r1 (A) Now we search the answer of the question that any -open set can be expressed as U = r1 (A) for some A 2 }(X). The answer is negative and it is followed by the following example. Example 8. Let X = fa; bg; I = f;; fagg and an ideal topological space and = f;; (...truncated)