Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 790 https://internationalpubls.com On G𝓖-Closed Sets in Grill Topological Spaces R. Anbarasan1,2 and M. Anitha3 1 Research Scholar (17231172091001) Affiliated to Manonmaniam Sundaranar University, Rani Anna Government College for Women, Tirunelveli, India 2 Assistant Professor, Department of Mathematics, PSN College of Engineering and Technology, Tirunelveli, India. Email: anbu.arasan1988@gmail.com 3 Associate Professor, Department of Mathematics Rani Anna Government College for Women, Tirunelveli, India Email: drmanitha10@gmail.com Article History: Received: 02-08-2024 Revised: 25-09-2024 Accepted: 20-10-2024 Abstract: In this article, we define a new class of gG-closed sets in a grill topological space and discuss the characterizations of gG-closed sets and gG-open sets by using the map s. Also analyze relationship between the gG-closed sets and some of the generalized closed sets. Keywords: gG-closed, gG-open, s-semiclosed, s-semi-dense. Mathematical Classification: 54A05, 54A10, 54D10. 1.Introduction Levine[14, 15] introduced the concepts of semiopen sets and generalized closed sets in topological spaces. Crossly et al.[9, 10] described the concepts of semi-closure and analysed the semi-topological properties. Chattopadhyay et al.[6, 7] described the metropic spaces and created the extensions of closure spaces. In [2, 5, 13, 24, 25, 26], studied the concepts of generalized closed sets through semiclosed sets. Choquet[8] defined the grill structure in topological spaces. Roy et al.[20, 21] developed the grill concepts and induced τ𝒢 topological space. In [1, 11, 16, 27], initiated different types of grill sets such as 𝒢-semiopen sets, 𝒢-- open sets and studied the decomposition of continuity via grill. Nasef[18], introduced s operator in grill topological space via semiopen sets and analyzed the essential topological characterizations. Mandal[17], generalized the closed sets in grill topological space and Saravanakumar et al.[22, 23] defined 𝒢sp -open sets and 𝒢sα -open sets through semiopen sets mailto:anbu.arasan1988@gmail.com mailto:drmanitha10@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 791 https://internationalpubls.com and characterized some topological structure. Anbarasan et al.[3, 4] studied generalized closed sets concepts via grill in generalized topological spaces. In this paper, we introduced new grill closed sets namely, g𝒢-closed in grill topological spaces. We characterized g𝒢-closed sets and g𝒢-open sets in grill topological spaces by use the mapping s and investigated some of their properties. We noticed that the idea of g𝒢-closed sets is a new generalization of 𝒢g-closed sets. Also, we analyzed relationship between this g𝒢- closed sets with existence generalized closed sets such as g-closed, s*g-closed, gs-closed, τ𝒢- closed, 𝒢g-closed etc. 2. Preliminaries In a topological space X, a subset A of X is said to be semiopen[14] (resp. -open[19], regular open[12]) if A  cl(int(A)) (resp. A  int(cl(int(A))), A = int(cl(A))). The complement X – A is called semiclosed (resp. -closed, regular closed). For a subset A of X, semiclosure of A defined as scl(A)[9] = {F  X : X – F is semiopen and A  F}; semiinterior of A defined as sint(A)[9] = {U  X : U is semiopen and A  U}. A subset A of X is said to be g-closed[15] (resp. s*g-closed[13], gs-closed[2]) if cl(A) ⊆ U (resp. cl(A)  U, scl(A) ⊆ U) whenever A  U and U is open (resp. U is semi-open, U is open) in X. A nonempty collection 𝒢 of subsets of a topological space (X, ) is called a grill[8] on X if (i)   𝒢, (ii) A  𝒢 and A  B implies that B  𝒢, (iii) A, B  X and A  B  𝒢 implies that A  𝒢 or B  𝒢. A triple (X , 𝒢) is called a grill topological space[20]. Let Y be a subset of X. Then 𝒢Y = {𝒢0  Y : 𝒢0  𝒢} is a grill on Y and grill topological subspace denoted by {Y, Y, 𝒢Y}. A mapping [20] (resp. s[18]) : P(X) → P(X) is defined by (A)[20] (resp. s(A)[18]) = {x  X: A  U  𝒢 for all U  (x) (resp. SO(X, x)} for all A  P(X), where (x) (resp. SO(X, x)) denotes the collection of all open (resp. semiopen) neighbourhoods of x. A mapping [20 (resp. s[18]) : P(X) → P(X) is defined by (A)[20] (resp. s(A)[18]) = A  (A) (resp. A  s(A)) for all A  P(X). Also  (resp. s) satisfies the Kuratowski closure axioms. Corresponding to a grill 𝒢 on a topological space (X, ), there exists a unique topology τ𝒢[20] (resp. τ𝒢 s [18]) = {U  X : (X – U) (resp. s(X – U)) = X – U}, where for any A  X, (A) (resp. s(A)) = A  (A) (resp. A  s(A)) = τ𝒢cl(A) (resp. τ𝒢 s cl(A))  cl(A) (resp. scl(A)) and   τ𝒢 (resp. SO(X)  τ𝒢 s ). A subset A of X is called (i) τ𝒢-closed[20] if Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 792 https://internationalpubls.com τ𝒢cl(A) = (A) = A  (A); -dense[20] if (A)  A; 𝒢g-closed[17] if (A)  U whenever A  U and U is open in X. Theorem 2.1.[18] Let (X, , 𝒢) be a grill topological space. Then, for every A, B  X, the following conditions are satisfied: (i) if A  B, then s(A)  s(B); (ii) s(A) = scl(s(A))  scl(A) and s(A) is semiclosed in X; (iii) s(s(A))  s(A); (iv) s(A  B) = s(A)  s(B); (v) if A  𝒢, then s(A) = . 3. g𝓖-Closed sets Definition 3.1. Let (X, , 𝒢) be a grill topological space and A be a subset of X. Then A is said to be g𝒢-closed if s(A)  U whenever A  U and U is open in X. The complement of a g𝒢-closed set is called a g𝒢-open set. Theorem 3.2. Let (X, , 𝒢) be a topological space and A be a subset of X. Then (i) If A is a τ𝒢-closed set, then A is 𝒢g-closed; (ii) If if A is a g-closed set, then A is 𝒢g-closed; (iii) If A is a 𝒢g-closed set, then A is g𝒢-closed; (iv) If A is a s*g-closed set, then A is g-closed; (v) If A is a g-closed set, then A is gs-closed; (vi) If A is a gs-closed set, then A is g𝒢-closed. Proof. (i) and (ii) Follows from the Remark 2.2.(b) and (d)[17]. (iii) Let A be a 𝒢g-closed set such that A  U and U  . Then by assumption, (A)  U. Since every open set is semiopen, we have that s(A)  (A). Therefore s(A)  U. Hence A is g𝒢-closed. (iv) Let A be a s*g-closed set such that A  U and U  . Since every open set is semiopen, A  U and U is semiopen. By assumption, cl(A) ⊆ U. Then by definition of g-closed, we have that A is g-closed. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 793 https://internationalpubls.com (v) Let A be a g-closed set such that A  U and U  . Then by assumption, cl(A)  U. Since every closed set is semiclosed, we have that scl(A)  cl(A). Therefore scl(A)  U. Hence A is gs-closed. (vi) Let A be a gs-closed set such that A  U and U  . Then by assumption, scl(A)  U. Since s(A)  scl(A). Therefore s(A)  U. Hence A is g𝒢-closed. Remark 3.3. The following examples shows that the reverse implication of above theorem is not true and the concepts of some generalizations of closed sets are independent. (i) Let X = {a, b, c, d},  = {, X, {a, b}, {a, b, d}} and 𝒢 = {X, {a}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}}. Then the set {a, c} is 𝒢g- closed (resp. g-closed), but not τ𝒢-closed (resp. s*g-closed). Also the set {d} is 𝒢g-closed (resp. gs-closed) but not g-closed. Moreover, the set {b, d} is g𝒢-closed but it is not 𝒢g-closed (resp. gs-closed). Here the set {b} is τ𝒢-closed, but not g-closed (resp. s*g-closed). As well as the set {a, c, d} is g-closed (resp. s*g-closed), but not τ𝒢-closed. (ii) Let X = {a, b, c, d},  = {, X, {a}, {b}, {a, b}} and 𝒢 = {X, {a}, {d}, {a, b}, {a, c}, {a, d}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}}. Then the set {a} is gs-closed but not 𝒢g-closed. The following diagram shows the relationship among various generalizations of closed sets. τ𝒢-closed 𝒢g-closed g𝒢-closed s*g-closed g-closed gs-closed Remark 3.4. In a grill topological space (X, , 𝒢), (i) every non-member of 𝒢 is g𝒢-closed; (ii) s is g𝒢-closed for every subset A of X; (iii) if 𝒢 = P(X) – {}, then s(A) = scl(A) and hence g𝒢-closed sets coincide with gs- closed sets. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 794 https://internationalpubls.com Theorem 3.5. Let (X, , 𝒢) be a grill topological space and A be a subset of X. Then the following statements are equivalent: (i) A is g𝒢-closed; (ii) scl(s(A))  U for every open set U containing A; (iii) for all x  scl(s(A)), cl({x})  A  ; (iv) scl(s(A)) – A contains no non empty closed set; (v) s(A) – A contains no non empty closed set. Proof. (i) ⇒ (ii). Let A be a g𝒢-closed set. Then clearly s(A) ⊆ U whenever A ⊆ U and U is open in X and so by Theorem 2.1, scl(s(A)) ⊆ U whenever A ⊆ U and U is open in X. This proves (ii). (ii) ⇒ (iii). Suppose x scl(s(A)). If cl({x}) ∩ A = , then A ⊆ X − cl({x}). By (ii), scl(s(A)) ⊆ X − cl({x}). This contradicts the fact that x ∈ scl(s(A)). Hence cl({x})  A ≠ . This proves (iii). (iii)  (i). Suppose that A is not g𝒢-closed. There exists an open set U such that A ⊆ U and s(A) is not contained in U. Then, there exists a point x ∈ s(A) such that x ∉ U. Then we have {x}  U =  and hence cl({x}) ∩ U = . Since A  U, cl({x})  A = . By Theorem 2.1, scl(s(A)) = s(A) and it follows that (iii) does not hold. Therefore, the proof completes. (iii)  (iv). Suppose F is a closed set of X contained in scl(s(A)) – A and x  F. Since F  A = , we have cl({x})  A = . Again, since x scl(s(A)), by (iii) we have cl({x})  A  , a contradiction. This proves (iv). It follows from Theorem 2.1 that (iv) and (v) are equivalent. Theorem 3.6. Let (X, , 𝒢) be a grill topological space and {Ai: i ∈ J} be a locally finite family of sets in X. Then i∈J( s(Ai)) = s(iJAi). Proof. Ai  iJAi implies s(Ai)  s(iJAi) for every i  J. This implies i∈J( s(Ai))  s(iJAi). Conversely, let x  s(iJAi) and V be any semiopen set of X containing x. Since {Ai: i ∈ J} is locally finite, there exists an open set U in X containing x that Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 795 https://internationalpubls.com intersects only a finite number of members, says, Ai1, Ai2, ..., Ain of {Ai: i ∈ J}. But x ∈ s(iJAi) implies (V ∩ U) ∩ (iJAi) = i∈J((V ∩ U) ∩ Ai)  𝒢 for every V  SO (X, x). This gives 𝑘=1 𝑛 ((V ∩ U) ∩ Aik)  𝒢 for every V  SO (X, x). Therefore, there exists at least one Aij ∈ {Ai1, Ai2, ..., Ain} such that (V ∩ U) ∩ Aij  𝒢 hence V ∩ Aij  𝒢. This gives x ∈ s(Aij) which implies x ∈ 𝑘=1 𝑛 (s(Aik)) and hence x ∈ i∈J( s(Ai)). This proves that s(iJAi)  i∈J( s(Ai)). This completes the proof. Theorem 3.7. Let (X, , 𝒢) be a grill topological space. If {Ai: i ∈ J} is a locally finite family of sets and each Ai is g𝒢-closed, then iJAi is g𝒢-closed in X. Proof. Let iJAi ⊆ U, where U is open in X. Since Ai is g𝒢-closed for each i ∈ J, then s(Ai) ⊆ U. Hence i∈J( s(Ai)) ⊆ U. By Theorem 3.6, s(iJAi) ⊆ U. Hence iJAi is g𝒢-closed in X. Remark 3.8. The following example shows that the intersection of two g𝒢-closed sets need not be g𝒢-closed. Let X = {a, b, c, d},  = {, X, {a, b}} and 𝒢 = {X, {a}, {b}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}}. Then A = {a, c} and B = {a, d} are g𝒢-closed sets, but A  B = {a} is not g𝒢-closed. Theorem 3.9. If A and B are subsets of a grill topological space (X, , 𝒢), then s (A  B) ⊆ s(A)  s(B). Theorem 3.10. Let (X, , 𝒢) be a grill topological space. If A is g𝒢-closed and B is closed in X, then A ∩ B is g𝒢-closed. Proof. Let U be an open set in X containing A ∩ B. Then A ⊆ U (X − B). Since A is g𝒢-closed, we have s(A) ⊆ U (X − B) and B ∩ s(A) ⊆ U. Using Theorem 3.9, s(A  B) ⊆ s(A)  s(B) ⊆ s(A) ∩ B ⊆ U because B is closed. This proves that A ∩ B is g𝒢-closed. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 796 https://internationalpubls.com Definition 3.11. A subset A of a grill topological space (X, , 𝒢) is said to be s-semiclosed if s(A) ⊆ A. Remark 3.12. Every 𝜏𝒢-closed set is s-semiclosed. The converse is not true. In example(ii) of Remark 3.3, A = {a} is s-semiclosed but it is 𝜏𝒢-closed. Definition 3.13. A subset A of a grill topological space (X, , 𝒢) is said to be s-semi- dense in-itself if A ⊆ s(A). Remark 3.14. Every s-semi dense in-itself set is -dense in-itself. Theorem 3.15. In a grill topological space (X, , 𝒢), a g𝒢-closed and s-semi-dense in- itself set is gs-closed. Proof. Suppose A is s-semi-dense in-itself and g𝒢-closed in X. Let U be any open set containing A. Since A is g𝒢-closed, s(A)  U and by Theorem 2.1, scl(s(A))  U. Since A is s-semi-dense in-itself, A  s(A) and hence scl(A)  U whenever A  U. This proves that A is gs-closed. Theorem 3.16. Let (X, , 𝒢) be a grill topological space and A be a g𝒢-closed subset of X. If B is a subset of X such that A  B  s(A), then B is g𝒢-closed. Proof. Let U be any open set of X such that B  U. Then A  U. Since A is g𝒢-closed, s(A)  U. By Theorem 2.1, we have s(B)  s(s(A))  s(A)  U and hence B is g𝒢-closed. Theorem 3.17. Let (X, , 𝒢) be a grill topological space and A  Y  X, where Y is α-open in X. Then s(A(𝒢𝑌, Y)) = s(A) ∩ Y. Proof. Assume that x ∈ X − (s(A) ∩ Y). Then either x ∈ Y or x ∉ Y Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 797 https://internationalpubls.com Case(i). x ∉ Y: Since s(A(𝒢𝑌, Y))  Y, then x ∉ s(A(𝒢𝑌, Y)). Case(ii). x ∈ Y: Since x ∉ s(A), there exists a semiopen set V in X containing x such that V ∩ A ∉ 𝒢. Since x ∈ Y and Y is α-open in X, we have a set Y ∩ V ∈ SO (Y, Y) such that x ∈ Y ∩ V and (Y ∩ V) ∩ A ∉ 𝒢 and hence (Y ∩ V) ∩ A ∉ 𝒢𝑌. Consequently, x ∉ s(A(𝒢𝑌, Y)). Hence, we get s(A(𝒢𝑌, Y))  s(A) ∩ Y. To prove the reverse implication, consider x ∉ s(A(𝒢𝑌, Y)). Then, for some semiopen set V in (Y, Y) containing x there exists U ∈ SO (X, x) such that V = U ∩ Y and we have (U ∩ Y) ∩ A ∉ 𝒢𝑌. Since A  Y, then U ∩ A ∉ 𝒢𝑌  𝒢 gives U ∩ A  𝒢 for some semiopen set U in (X, τ) containing x. This proves x ∈∉ s(A). This completes the proof. Theorem 3.18. Let (X, , 𝒢) be a grill topological space and A  Y  X. If A is g𝒢- closed in (Y, Y, 𝒢𝑌) and X is α-open and s-semiclosed in X, then A is g𝒢-closed in X. Proof. Let A  U and U be open in X. Then s(A(𝒢𝑌, Y)) = s(A) ∩ Y  U ∩Y. Then we have Y  U (X – s(A)). Since Y is s-semiclosed, we have s(A)  s(Y)  Y  U (X – s(A)). This proves that s(A)  U. This completes the proof. Theorem 3.19. Let (X, , 𝒢) be a grill topological space and A  Y  X. If A is g𝒢- closed in X and Y ∈ , then A is g𝒢-closed in (Y, Y, 𝒢𝑌). Proof. Let U be an open subset of (Y, Y) such that A  U. Since Y ∈ , then U ∈ . Thus s(A)  U. By Theorem 3.17, s(A(𝒢𝑌, Y)) = s(A) ∩ Y  U ∩ Y= U and we have s(A(𝒢𝑌, Y))  U. Hence A is g𝒢-closed in (Y, Y, 𝒢𝑌). Corollary 3.19. Let (X, , 𝒢) be a grill topological space and A  Y  X, where Y is a regular open subset of X. Then A is g𝒢-closed in (Y, Y, 𝒢𝑌) if and only if A is g𝒢- closed in X. Theorem 3.20. Let (X, , 𝒢) be a grill topological space and A  X. If A is g𝒢-closed, then A ∪ (X − s(A)) is g𝒢-closed. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 798 https://internationalpubls.com Proof. Suppose A is g𝒢-closed. Let U be an open set such that A ∪ (X – s(A))  U. Then X − U  X − (A ∪ (X – s(A))) = s(A) − A. Since A is g𝒢-closed, by Theorem 3.5, it follows that s(A) − A contains no non-empty closed set. This implies X − U =  or X = U. Thus, X is the only open set containing A ∪ (X − s(A)). This gives s(A  (X – s(A)))  X. This proves A ∪ (X − s(A)) is g𝒢-closed. Theorem 3.21. Let (X, , 𝒢) be a grill topological space. Then A  (X – s(A)) is g𝒢- closed if and only if s(A) – A is g𝒢-open. Proof. 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