Ratio Mathematica Volume 44, 2022 Semi generalization of δI*-closed sets in ideal topological space Dr K Palani 1 M Karthigai Jothi2 Abstract In this paper we introduce the notion of semi generalized I*-closed sets or gsI*- closed sets using semi open sets and investigate its basic properties and characterizations in an ideal topological space. This class of sets is properly lies between the class of I*-closed sets and the class of g-closed sets. Also, study the relationship with various existing closed sets in ideal topological spaces. Moreover, we introduce and study the concept of maximal gsI*-closed sets. Keywords: ideal topological space, I*-closed sets, gsI*-closed sets. 2010AMS subject classification: 05C693 1Associate Professor and Head, PG & Research Department of Mathematics, A.P.C Mahalaxmi College for Women, Thoothukudi-2. Tamilnadu, India.E-mail: palani@apcmcollege.ac.in 2Research Scholar, Reg. No: 21212012092005, PG & Research Department of Mathematics, A.P.C Mahalaxmi College for Women, Thoothukudi-2. Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli -12. Tamilnadu, India.E-mail: jothiperiyasamy05@gmail.com. 3Received on June 10 th, 2022. Accepted on Sep 1st, 2022. Published on Nov 30th, 2022. doi: 10.23755/rm.v44i0.925. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY license agreement. 354 mailto:palani@apcmcollege.ac.in mailto:jothiperiyasamy05@gmail.com Dr K Palani M Karthigai Jothi 1. Introduction and Preliminaries An ideal I is a non-empty collection of subsets of X which satisfies: (i) A∈I and B ⊆ A implies B ∈I, and (ii) A∈I and B ∈I implies A ∪ B ∈I. Given a topological space (X, τ) with an ideal I on X called ideal topological space denoted by (X, τ, I). Kuratowski [5] and vaidhyanathaswamy [18] was studied the notion of ideal topological spaces, J. Dontchev, M. Ganster [3], Navaneethakrishnan, P. Paulraj Joseph [13], D. Jankovic, T. R. Hamlett [4], M. N. Mukherjee, R. Bishwambhar, R. Sen [10], A. A. Nasef, R. A. Mahmond [12] etc., were investigated applications to various fields of ideal topology. If P(X) is the collection of all subsets of X a set operator (.)*: P(X) → P(X) called a local function [5] for any subset A of X with respect to I and τ is defined as, A*(I, τ) = {x ∈X: U ∩ A ∉ I for every U ∈ τ(x)}, where τ(x) = {U ∈ τ / x ∈ U}. A kuratowski closure operator cl*(A) for a topology τ*(I, τ) called *- topology finer than τ is defined by cl*(A) = A ∪A*(I, τ). A subset A of X is said to be δ-closed [19] set if clδ(A) = A, where clδ(A) = {x X: Int(cl(U)) ∩ A , for every U (x)}. The complement of δ-closed set is δ-open set. A subset A subset A of a space (X, ) is an -open [14] (resp. semi open [7]) set if A int(cl(int(A))) (resp. A  cl(int(A))). The complement of a semi open (resp.-open) set is called a semi closed (resp.-open). Definition 1.1. Let (X,) be a topological space. A subset A of X is said to be (i) a generalized closed (briefly, g-closed) set [6] if cl(A)  U whenever A  U and U is open in (X, ). (ii) a generalized semi closed (briefly, gs-closed) set [1] if scl(A)  U whenever A  U and U is open set in (X, ). (iii) a semi-generalized closed (briefly, sg-closed) set [2] if scl(A)  U whenever A  U and U is semi open set in (X, ). (iv)an-generalized closed (briefly,g-closed) se [8]t if cl(A)  U whenever A  U and U is open in (X, ). (v) a generalized -closed (briefly, g-closed) set [9] if cl(A)  U whenever A  U and U is -open in (X, ). (vi) a ĝ (or) w-closed set [20] if cl(A)  U whenever A  U and U is semi open set in (X, ). Definition 1.2. [21] Let (X, , I) be an ideal topological space. A subset A of X is said to be an Ig-closed set if A*  U whenever A  U and U is open in X. Definition 1.3. [21] Let (X, , I) be an ideal topological space, A a subset of X and x is a point of X. Then 355 Semi generalization of δI*-closed sets in ideal topological space (1) x is called a -I-cluster point of A if A ∩int(cl*(U)) ≠ , for each open neighborhood U of x. (2) the family of all -I-cluster points of A is called the -I-closure of A and is denoted by [A]  -I. (3) a subset A is said to be -I-closed if [A]-I = A. The complement of a -I-closed set of X is said to be -I-open. Lemma 1.4. [21] Let A and B be subsets of an ideal topological space (X, , I). Then, the following properties hold. (1) A  [A]-I. (2) If A  B, then [A]-I [B]-I. (3) [A]-I = ∩ {F  X / A  F and F is -I-closed}. (4) If A is -I-closed set of Xs for each , then ∩ {A / } is -I-closed. (5) [A]-I is -I-closed. Lemma 1.5. [21] Let (X, τ, I) be an ideal topological space and τ-I = {A  X / A is -I- open subset of (X, τ, I)}. Then τ-I is a topology such that τSτ-I τ, where τS is the collection of -open sets. Definition 1.6. [16] Let (X, τ, I) be an ideal topological space and A a subset of X. Then [A]*(I, τ) = {x ∈X: int[U]δ-I∩ A  for every U ∈τ(x)} is called local δI-closure function of A with respect to the ideal I and topology τ, where τ(x) = {U ∈ τ / x ∈ U}. A subset A is said to be δI-closed if [A]* = A. The complement of δI-closed set is called δI-open set. Remark 1.7.[16] Always, (i) [A]* is closed, (ii) []* =  and [X]* = X, (iii) A ⊆ [A]*. Lemma.1.8. [16] Let (X, τ, I) be an ideal topological space and A, B subsets of X. Then for local δI-closure functions the following properties hold. (i) If A ⊆B then [A]*⊆[B]*. (ii) [A ∪B] *= [A]* ∪ [B]*. (iii) [A ∩ B] * ⊆[A]* ∩ [B]*. (iv) [[A]*] * = [A]*. Lemma 1.9.[16] (i) cl(A) ⊆ [A]*, (ii) A* ⊆ [A]*, (iii) clδ(A) ⊆ [A]*, (iv) [A]δ-I⊆ [A]*. 356 Dr K Palani M Karthigai Jothi Definition 1.10. [17] A subset A of an ideal space (X, , I) is called gδI*-closed if [A]* ⊆ U whenever A ⊆ U and U is open in (X, , I). The complement of a gδI*-closed set in (X, , I) is called gδI*open set in (X, , I). 2. gsI*- closed Sets In this section we introduce gsI*-closed sets and discuss the relationship with some existing sets. Definition 2.1. A subset A of an ideal topological space (X, , I) is called gsI*-closed if [A]*⊆ U whenever A ⊆ U and U is semi open set in (X, , I). The complement of gsI*-closed set in (X, , I), is called gsI*-open set in (X, , I). Theorem 2.2. EveryI*-closed set is gsI*-closed. Proof. Let A be any I*-closed set and U be any semi open set containing A. Since A is I*-closed, [A]* = A. Therefore, A is gsI*-closed set in (X, , I). Remark 2.3. The converse of the above Theorem 2.2 is need not be true as shown in the following Example 2.4. Example 2.4. Let X = {a, b, c},  = {X, , {b}, {c, d}, {b, c}, {b, c, d}}, I = {, {d}}. Let A = {a, b, c}. Then, A is gsδI*-closed but not δI*-closed. Theorem 2.5. In an ideal topological space (X, , I), every gsI*-closed set is (i) ĝ -closed set in (X, ). (ii) g-closed (resp. gα, αg, sg, gs) -closed set in (X, ). (iii) Ig -closed set in (X, , I). Proof. (i) Let A be a gsI*-closed set and U be any semi open set in (X, , I) containing A. Since A is gsI*-closed, [A]*⊆ U. Then cl(A) ⊆ U and hence A is ĝ -closed in (X, , I), by Lemma 1.9. (ii) By [20], every ĝ-closed set is g-closed (resp. gα-closed, αg-closed, sg-closed, gs- closed) set in (X, , I). Therefore, it holds. (iii) Since every g-closed set is Ig-closed, it holds. Remark 2.6. The following Example 2.7 shows that, the converse of the above Theorem 2.5 (i) is not always true. Example 2.7. Let X = {a, b, c, d},  = {X, , {b}, {a, b}, {b, c}, {a, b, c}, {a, b, d}} and I = {, {b}}. Let A = {c, d}. Then A is ĝ-closed set but not gsI*-closed. Remark 2.8. The following Examples shows that, the converse of Theorem 2.5 (ii) is not true. 357 Semi generalization of δI*-closed sets in ideal topological space Example 2.9. Let X = {a, b, c, d},  = {X, , {b}, {c}, {b, c,}} and I = {, {d}}. Let A = {d}. Then A is g-closed, g-closed, g-closed but not gsI*-closed. Example 2.10. Let X = {a, b, c, d},  = {X, , {a}, {c, d}, {a, c, d}, {b, c, d}} and I = {, {a}}. Let A = {a, b}. Then A is gs-closed and sg-closed but not gsI* closed. Remark 2.11. The following Example 2.12 shows that, the converse of Theorem 2.5 (iii) is not always true. Example 2.12. Let X = {a, b, c, d},  = {X, , {b}, {a, b}, {b, c}, {a, b, c}, {a, b, d}} and I = {, {b}}. Let A = {b}. Then A is Ig -closed but not gsI*-closed. 3. Characterizations In this section we study some of the basic properties and characterizations of gsI*- closed sets. Theorem 3.1. Let (X, , I) be an ideal space and A a subset of X. Then [A]* is semi closed. Proof. By Remark 1.7, [A]* is closed and hence it is semi closed. Theorem 3.2. Let (X, , I) be an ideal space and A ⊆ X. If A ⊆ B ⊆[A]*, then [A]* = [B]*. Proof. Since A ⊆ B, [A]*⊆[B]* and since B ⊆[A]*, [B]*⊆[[A]*] * = [A]*, By Lemma 1.8 and Lemma 1.9. Therefore, [A]* = [B]*. Theorem 3.3. Let (X, , I) be an ideal space. Then [A]* is always gsI*-closed for every subset A of X. Proof. Let [A]*⊆ U, where U is semi open. Always, [[A]*] * = [A]*. Hence [A]* is gsI*-closed. Theorem 3.4. Let (X, , I) be an ideal space and A ⊆ X. If sker(A) is gsI*-closed, then A is also gsI*-closed. Proof. Suppose that, sker(A) is a gsI*-closed set. If A ⊆ U and U is semi open, then sker(A) ⊆ U. Since sker(A) is gsI*-closed, [sker(A)]*⊆ U. Always, [A]*⊆[sker(A)]*. Thus, A is gsI*-closed. The following Example 3.5 shows that, the converse of the above Theorem 3.4 is not always hold. Example 3.5. In Example 2.12, let A = {a, b}. Then A is gsI*-closed. But, sker(A) = {a ,b, c} is not gsI*-closed. 358 Dr K Palani M Karthigai Jothi Theorem 3.6. If A is gsI*-closed subset in (X, , I), then [A]* – A does not contain any nonempty closed set in (X, , I). Proof. Let F be any closed set in (X, , I) such that F ⊆[A]* – A then A ⊆ X – F and X – F is open and hence semiopen in (X, , I). Since A is gsI*-closed, [A]*⊆X – F. Hence, F ⊆ X – [A]*. Therefore, F ⊆ ([A]* – A)  (X – [A]*) = . Remark 3.7. The converse of the above Theorem 3.6 is not always true as shown in the following Example 3.8. Example 3.8. Let X = a, b, c,  = X,, a, {b}, {a, b} and I = , {c}, {d}, {c, d}. Let A = a, b, c. Then [A]* – A = X – a, b, c = d does not contain any nonempty closed set. But A is not a gsI*-closed subset of (X, , I). Theorem 3.9. For a subset A of an ideal space (X, , I), cl(A) – A is gsI*-closed if and only if A  (X – cl(A)) is gsI*-open. Proof. Necessity - Let F = cl(A) – A. By hypothesis, F is gsI*-closed and X – F = X  (X – F) = X  (X – (cl(A) – A)) = A  (X – cl(A)). Since X – F is gsI*-open, A (X– cl(A)) is gsI*-open. Sufficiency-Let U = A  (X – cl(A)). By hypothesis, U is gsI*-open. Then X – U is gsI*-closed and X – U = X – (A  (X – cl(A))) = cl(A)  (X – A) = cl(A) – A. Hence proved. Theorem 3.10. Let (X, , I) be an ideal space. Then every subset of X is gsI*-closed if and only if every semiopen subset of X is I*-closed. Proof. Necessity - Suppose every subset of X is gsI*-closed. If U is a semiopen subset of X, then U is gsI*-closed and so [U]* = U. Hence, U is I*-closed. Sufficiency - Suppose A⊆ U and U is semiopen. By hypothesis, U is I*-closed. Therefore, [A]*⊆[U]* = U and hence A is gsI*-closed. Theorem 3.11. Let (X, , I) be an ideal space. If every subset of X is gsI*-closed, then every open subset of X is I*-closed. Proof. Suppose every subset of X is gsI*-closed. If U is an open subset of X, then U is gsI*-closed and so [U]*⊆ U, since every open set is semiopen. Hence, U is I*-closed. Theorem 3.12. Intersection of a gsI*-closed set and aI*-closed set is always gsI*- closed. Proof. Let A be a gsI*-closed set and G be any I*-closed set of an ideal space (X, , I). Suppose A  G ⊆ U and U is semiopen set in X. Then, A⊆ U  (X – G). Now, X – G is I*-open and hence open and so semiopen set. Therefore, U  (X – G) is a semiopen set containing A. But A is gsI*-closed and therefore, [A]*⊆ U (X – G). 359 Semi generalization of δI*-closed sets in ideal topological space Therefore, [A]* G ⊆ U which implies that, [A  G] *⊆ U. Hence, A  G is gsI*- closed. Theorem3.13. In an ideal space (X, , I), for each x X, either x is semiclosed or xc is gsI*-closed. Proof. Suppose that x is not a semiclosed set, then xc is not a semiopen set and hence X is the only semiopen set containing xc. Therefore, [xc] *⊆ X and hence xc is gsI*-closed in (X, , I). Theorem 3.14. Every gsI*-closed, semiopen set is I*-closed. Proof. Let A be a gsI*-closed, semiopen set in (X, , I). Since A is semiopen such that A ⊆ A, by hypothesis, [A]*⊆ A. Thus, A is I*-closed. Corollary 3.15. Every gsI*-closed; open set is I*-closed set. Theorem 3.16. If A and B are gsI*-closed sets in an ideal topological space (X, , I), then A  B is a gsI*-closed set in (X, , I). Proof. Suppose that A  B ⊆ U, where U is semi open set in (X, , I). Then A⊆ U and B ⊆ U. Since A and B are gsI*-closed sets in (X, , I), [A]*⊆ U and [B]*⊆ U. Always, [A  B] * = [A]*[B]*. Therefore, [A  B] *⊆ U, whenever U is semi open. Hence, A  B is gsI*-closed set in (X, , I). Theorem 3.17. Let (X, , I) be an ideal space. If A is a gsI*-closed subset of X and A ⊆ B ⊆[A]*, then B is also gsI*-closed. Proof. The proof is clear. Theorem 3.18. A subset A of an ideal space (X, , I) is gsI*-closed if and only if [A]*⊆ sker(A). Proof. Necessity - Suppose A is gsI*-closed and x [A]*. If x  sker(A), then there exist a semiopen set U such that A ⊆ U but x  U. Since A is gsI*-closed, [A]*⊆ U and so x [A]*, a contradiction. Therefore, [A]* sker(A). Sufficiency - Suppose that [A]*⊆ sker(A). If A ⊆ U and U is semiopen then sker(A) ⊆ U and so [A]*⊆ U. Therefore, A is gsI*-closed. Theorem3.19. Let A be a semi - set of an ideal space (X, , I). Then A is gsI*-closed if and only if A is I*-closed. Proof. Necessity - Suppose A is gsI*-closed. Then by Theorem 3.18, [A]*⊆sker(A) = A, since A is semi - set. Therefore, A is I*-closed. Sufficiency - The proof is follows from the Theorem 2.2. 360 Dr K Palani M Karthigai Jothi Definition 3.20. A proper nonempty gsI*-closed subset A of an ideal space (X, , I) is said to be maximal gsI*-closed if any gsI*-closed set containing A is either X or A. Example 3.21. Let X = {a, b, c, d},  = {X, , {b}, {c, d}, {b, c, d}} and I = {, {d}}. Then {a, b, c} is a maximal gsI*-closed set. Theorem 3.22. In an ideal space (X, , I), the following are true. (i) Let F be a maximal gsI*-closed set and G be a gsI*-closed set. Then F  G = X or G ⊆F. (ii) Let F and G be maximal gsI*-closed sets. Then F  G = X or F = G. Proof. (i)Let F be a maximal gsI*-closed set and G be a gsI*-closed set. If F  G = X, then there is nothing to prove. Assume that, F  G ≠ X. Now, F ⊆ F  G. By Theorem 3.16, F  G is a gsI*-closed set. Since F is maximal gsI*-closed, we have F  G = X or F  G = F. Hence, F  G = F and so G ⊆ F. (ii) Let F and G be maximal gsI*-closed sets. If F  G = X, then there is nothing to prove. Assume that, F  G ≠ X. Then by (i), F ⊆ G and G ⊆ F, which implies that, F = G. Theorem 3.23. A subset A of an ideal space (X, , I) is gsI*-open if and only if F ⊆[A]int* whenever F is semiclosed andF ⊆ A. Proof. Necessity - Suppose A is gsI*-open and F be a semiclosed set contained in A. Then X – A ⊆ X – F and hence [X – A]*⊆ X – F. Thus, F ⊆ X – [X – A]* = [A]int*. Sufficiency - Suppose X – A ⊆ U, where U is semiopen. Then X – U ⊆ A and X – U is semiclosed. Then X – U ⊆[A]int*, which implies [X – A]*⊆U. Therefore, X – A is gsI*-closed and hence A is gsI*-open. Theorem 3.24. If A is a gsI*-open subset of an ideal space (X, , I) and [A]int*⊆ B ⊆A. Then B is also a gsI*-open subset of (X, , I). Proof. Suppose F ⊆ B, where F is semiclosed set. Then, F ⊆ A. Since A is gsI*-open, F ⊆[A]int*. Since [A]int*⊆[B]int*, we have F ⊆[B]int*. By the above Theorem 3.23, B is gsI*-open. References [1] S. P. Arya, T. Nour, Characterizations of S-normal Spaces, Indian J. Pure Appl. Math., 21 (8), 717 - 719. 1990. [2] P. Bhattacharya, B. K. Lahiri, Semi-generalized Closed Sets in Topology, Indian J. Math., 29, 375 – 38. 1987. [3] J. Dontchev, M. Ganster., D. Rose, Ideal Resolvability. Topology and its Appl., 93, pp.1-16. 1999. 361 Semi generalization of δI*-closed sets in ideal topological space [4] D. Jonkovic, T.R. Hamlett, New Topologies from old via Ideals, Amer. Math., Monthly 97, pp. 295-310. 1990. [5] K. Kuratowski, Topology, Vol. I. New York: Academic Press, 1996. [6] N. Levine, Generalized Closed Sets in Topology, Rend. Circ. Mat. Palermo., 19, 89 - 96. 1970. [7] N. Levine, Semiopen Sets and Semi continuity in Topological Spaces, Amer. Math. Monthly,70, 36 - 41.1963. [8] H. Maki, R. Devi and K. Balachandran, Generalized  -closed sets in Topology, Bull. Fukuoka Uni., Ed part III, 13 - 21. 1993. [9] H. Maki, R. Devi and K. Balachandran, Associated Topologies of Generalized  - closed Sets and  -generalized Closed Sets, Mem. Fac. Sci. Kochi Univ. Ser. A. Math., 15, 57 – 63. 1994. [10] M.N. Mukherjee, R. Bishwambhar, R. Sen, On Extension of Topological Spaces in terms of Ideals. Topology and its Appl., 154, pp. 3167-3172, 2007. [11] B.M. Munshi and D. S. Bassan, Superc continuous Mappings, Indian J. Pure Appl. Math., 13, 229 – 236, 1982. [12] A.A. Nasef, Rearmament, Some Applications via Fuzzy ideals. Chaos, Solitons and Fractals 13, pp. 825-831, 2002. [13] M. Navaneethakrishnan, J. Paulraj Joseph, g-closed sets in ideal Topological Spaces, Acta. Math. Hungar., DOI.10.107/s10474-007-7050-1. [14] O. Njastad, On Some Classes of Nearly Open Sets, Pacific J. Math., 15 (3), 961 – 970. 1965. [15] T. Noiri, On -continuous Functions, J. Korean Math. Soc., 16, pp 161 – 166, 1980. [16] K. Palani, Karthigaijothi, δI*-Closed sets in Ideal Topological Spaces- Chap. II- Ph.d –Mini Project. [17] K. Palani, Karthigaijothi, Generalization of δI*-Closed Sets in Ideal Topological Spaces – Ph.d Mini Project – Chap. III. [18] V. Vaidyanathaswamy, The Localization Theory in set Topology, Proc. Indian. Acad. Sci. 20, 1945. [19] N.V. Velicko, H-Closed Topological Spaces, Math. Sb.,70, pp. 98-112, 1996. [20] M.K.R.S. Veerakumar, On ĝ –closed sets in Topological Spaces, Bull.Allh.Math. Soc., 99 – 112. 2003. [21] S. Yuksel, A. Acikgoz, T. Noiri, On δ-I-Continuous Functions, Turk J Math, 29, pp.39-51, 2005. 362