Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 740 https://internationalpubls.com On δgα Closure and δgα Interior S In TSS M. Madhesan1, V.E. Sasikala2,* 1Research Scholar, Vels Institute of Science Technology and Advanced Studies, Chennai, Tamil Nadu, India. 2,*Research Supervisor, Assistant Professor, Vels Institute of Science Technology and Advanced Studies, Chennai, Tamil Nadu, India 2,* Corresponding author: sasikala.sbs@velsuniv.ac.in Article History: Received: 21-04-2024 Revised: 11-06-2024 Accepted: 24-06-2024 Abstract: The purpose of this research article is to explain the new notions of δgα - derived, δgα- closure, δgα-interior, δgα-nbd., and moreover, the connections among them are identified. Keywords: δgα-O, δgα-closure, δgα-interior, δgα-nbd., δgα-Derived, δgα-border, δgα- frontier, δgα-exterior and δgα-saturated. 1. Introduction In many application domains, like data mining, the significance of general TSs is growing quickly [13]. Mathematizing both quantitative and qualitative data is possible with topological structures on the data collection serving as appropriate mathematical models. Nowadays, a large number of topologists worldwide are studying generalized Os because they are crucial to general topology. A widely recognized concept that serves as a source of inspiration is the idea of αδ-O [12], which was first presented by R. Devi et al., We shall carry out the analysis of related functions with αδ-O and αδ- C s in this research. We present and define the terms "αδ-D," "αδ-exterior," as well as deduce their relationship. Furthermore, we present a brand-new function known as αδ-Totally-Continuous Functions. Additionally, as delineated and examined in these works by D. Sivaraj et al., [1-4], A Study on Beta Generalized C s in TS, Soft α–O s, [19–25] On soft regular star generalized star C s in soft TSs and [5-10] semi-closure, a note on soft g-C s Hildebrand S. K. et al., Regarding very αδ super irresolute functions in TSs, Benchalli S et al., On RW-C s in TSs, [11–12] V. Kokilavani et al., the αδ- kernal and αδ-closure via αδ-O s in TSs, D–αδ -s and related separation axioms in TSs, [15–18] Davis A. S., In addition, the fundamental characteristics of these functions as well as TS preservation theorems are presented and examined. 2. Preliminaries Let X be a TS and A be X's subset. A's interior and closure are represented, respectively, by the symbols cl(A) and int(A). Definition 2.1: A sub A of a space (X, ) is called (1) Regular-O [15] if A = int(cl(A)). (2) semi-O [15] if A ⊆ cl(int(A)). (3) α-O [2] if A ⊆ int(cl(int(A))). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 741 https://internationalpubls.com (4) δ-semi-O [12] A ⊆ cl(Int δ (A)). Levine's g-Cs have been compared to other generalized closure operators or classes of generalized-O s to generate a variety of ideas. A useful tool for characterizing TSs is the generalize-C. The union of all regular Os of X contained in A is the δ-interior [10] of a sub A of X, and it is represented by Intδ(A). If A = Intδ(A), then the sub A is referred to as δ-O [10]. That is, if an is the union of regular-O s, then it is δ-O. A δ-C is the complement of a δ-O. Detailed study in this regard by many investigators has enriched the field of generalized C s to a considerable extent. Nbd. is one of the core concepts of topology. Nbd. in topology have significant role in the applications of mathematics. 3. δgα −Closure (δgα −cl) and δgα −Interior (δgα −int) In TSS In this paper we establish the notion of δgα − closure, δgα − interior in the TSs. Definition 3.1. A subset M of a TS X is called a δgα -O (briefly, δgα-O) if Mc is δgα -C. The family of all δgα - O s in a TS X is represented by δgα-O(X). Example 3.2. Let X= {e, f, g, h }, τ = {X, φ, {e}, {f}, {e, f}, {e, g}, {e, h}, {e, f, g}, {e, f, h}, {e, g, h}} then the δgα C s are {X, φ, {e}, {f}, {g}, {e, f, g}} and δgα -O s are { X, 𝜑, {f, g, h}, {e, g, h},{e, f, h} {h}}. Definition 3.3. The δgα − cl of a subset A of (X, ) is denoted by δgα− cl(A) and is defined as the intersection of all δgα − C s containing A and is denoted by δgα-cl(A). δgα-cl(A) is the smallest δgα-C containing A. Therefore, δgα − cl(A)= {M  X: A M and M is δgα − C}. Definition 3.4. The δgα − int of subset A of (X, ) is denoted by δgα −int(A) and is defined as the union of all δgα − O contained in A and is denoted by δgα-int(A). δgα-int(A) is the largest δgα O sub of A. Therefore, δgα-int(A) = {N  X: N  A and N is δgα − O}. Remark 3.5. (i). Every O is δgα -O. (ii). Finite intersection of δgα-O s need not be δgα-O. (iii). Finite union of δgα-O s need not be δgα-O. Theorem 3.6. A subset M of a space Z is δgα -O ⟺ F⊆ αint(M) whenever F⊆M where F is δ-C. Proof: Let M be a δgα -O subset of X and suppose F⊆ M where F is δ-C. Then Z-M is δgα-C and Z- M ⊆ Z-F where Z-F is δ-O in Z. By Definition of δgα -C, αcl(Z-M)⊆ Z-F. Since αcl (Z - M) = Z - αint(M), then Z - αint(M) ⊆ Z - F. Therefore F ⊆ α int(M). Conversely, let F ⊆ α int(M) be true whenever F ⊆ M and F is δ-C in Z, then Z - α int(M) ⊆Z - F. That is, αcl(Z - M) ⊆ Z - F. Thus Z - M is δgα – C and M is δgα -O. Theorem 3.7. If F is δgα-O sub of a space Z whereas αint(F) ⊆G⊆F, then V is δgα -O. Proof: From the Definition 3.1 and δgα – C. Theorem 3.8. If S is any δgα -O sub of a space X whereas αint(S) ⊆N, then S⋂N is δgα -O. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 742 https://internationalpubls.com Proof: Let S be any δgα-O sub of X and αint(S) ⊆ N, then S ⋂ αint(S) ⊆ S ⋂ N ⊆ S. Since αint(S) ⊆ S, then αint(S) ⊆ S ⋂ N ⊆ S and from Theorem 3.5, S ⋂ N is δgα-O in X. Theorem 3.9. Let M be any δgα-C subset. Then αcl(M)-M is δgα -O. Proof: Let M be a δgα-C and F be a δ-C in X whereas F ⊆ αcl(M) – M, Then by Theorem M be a δgα -C sub of a space X, then αcl(M)-M contains no non empty δ-C ., F = ∅ and hence F ⊆αint(αcl(M) - M). Therefore, by Theorem 3.4, αcl(M) - M is δgα -O in X. Lemma 3.10. Let Y be a δgα-subspace of X. If U is δgα O in Y and Y is δgα O, then U is δgα-O. Proof: Given U is δgα O in Y, U = Y ⋂ G for some G δgα O in X. But Y and G are both δgα O in X so Y ⋂G is also δgα O in X. Theorem 3.11. Assume that M and N be any two subs of a TS. Then the succeeding properties hold. 1. E is δgα − C iff δgα − cl(E) = E. 2. δgα − cl(E) is the smallest δgα − C sub of X containing E. 3. δgα − cl () is empty, δgα − cl (X)= X. 4. δgα − cl(E) is a δgα − C in (X, ). 5. If E  F, then δgα − cl(E)  δgα − cl(F) 6. δgα − cl (E F) = δgα − cl(E)  δgα − cl(F). 7. δgα − cl (E F) = δgα − cl(E)  δgα − cl(F). 8. δgα − cl (δgα − cl(E)) = δgα − cl(F). Proof: 1. For any sub E of X we have E  δgα − cl(E). Assume that E is a δgα − C in (X, ). But E  E. Also E {H  X: E  H and E is δgα − C}, it gives E =  {H  X : E  H and H is δgα −C} E. Then δgα − cl(E)  E. So  = δgα −cl(E). 2. Beginning the definition of δgα − cl, δgα − cl(E) is C. Suppose if F is any δgα − C then δgα − cl(E)  F. Hence δgα − cl(E) is the smallest δgα − C in (X, ) containing . 3. Proof is obvious from the definition. 4. Proof is apparent from the definition. 5. If E  F then E  δgα − cl(F) because F  δgα − cl(F) for all F. Hence δgα − cl(F) is the δgα − C containing E. But δgα −cl(E) is smallest δgα − C containing E. So δgα − cl(E)  δgα − cl(F). 6. We know the result E  (E F) and N  E  F, from the above result, δgα − cl(E)  δgα − cl (E F) also δgα − cl(F)  δgα − cl (E F) and δgα − cl(F)  δgα − cl ( F). So δgα − cl(E)  δgα − cl(F)  δgα − cl (E  F). But δgα − cl(E) is δgα − C containing E and δgα − cl(F) is δgα − C containing F. Hence δgα − cl(E)  δgα − cl(E) is δgα − C containing E  F. Here δgα − cl (E  F) is Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 743 https://internationalpubls.com δgα − C containing (E  F). Therefore δgα − cl ()  δgα − cl(F)  δgα − cl(E  F). Therefore we get δgα − cl(E  F) = δgα − cl(E)  δgα − cl(F). 7. We know that (E  F)  E and (E  F)  F. By (v) δgα − cl(E  F)  δgα − cl(E) and δgα − cl(E  F)  δgα − cl(E)  δgα − cl(F). 8. δgα − cl(E) is a δgα - C in (X, ). Let then K is δgα -C δgα − cl(E) = K, in (X, ). Using (i) δgα − cl(K) = K, which gives δgα − cl (δgα − cl(E)) = δgα − cl(E). Remark 3.12. For any sub A  X, 1. δgα −int(E) is the largest δgα − O ⊆ E. 2. A is δgα − O, iff δgα −int(A) = A. 3. δgα − int(X) = X. 4. δgα − int() = . 4. δgα - NBD In TSS: In this paper we establish the notion of δgα − nbd. in the TSs. Definition 4.1. Let N be a sub of TS (X, ), then N is said to be δgα − nbd. of point x X if there exist a δgα − O (G) where as xG  N. The group of all δgα − nbd. of an element x X called δgα − nbd. of x and is signified by δgα −(x). Example 4.2. Let X= {e, f, g, h }, τ = {X, φ, {e}, {f}, {e, f}, {e, g}, {e, h}, {e, f, g}, {e, f, d}, {e, g, h}} then the δgα C s are {X, φ, {e}, {f}, {g}, {e, f, g}} and δgα - O s are { X, 𝜑, {f, g, h}, {e, g, h},{e, f, h} {h}}. Let b X, if there exist a δgα − O G whereas fG  N, then δgα-nbd. of an element b X, That is δgα −(f) ={X, 𝜑, {f, g, h}, {e, f, h}}. Theorem 4.3. A sub P of (X, ) is δgα − C and p δgα − cl(P) iff Y  P is not empty for any δgα − nbd. Y of p in (X, ). Proof: Assume p is not an element of δgα − cl(P). Then there exits δgα −C E of X whereas P  E and p is not an element of E. Hence p(X \ E) is δgα − O in X. But P (X \ E) is empty. This is a contradiction. Thus p δgα − cl(P). Conversely assume that there is a δgα − nbd. Y of a pt. p X where as Y  P is empty. Then there is a δgα − O E of X whereas p E  Y. Hence E  P is empty, p(X / E). So δgα − (X \ E) and p is not an element of δgα − cl(P). This is a contradiction to p δgα − cl(P). Thus, the intersection of Y and P is not empty. Theorem 4.4. If B is δgα − O then it is δgα − nbd. of each of its pts. Proof: Consider a δgα − O of (X,). Then by definition for all bB, b  . So M is δgα − nbd. of each of its pts. Theorem 4.5. If B X is a δgα − C, bBc, then there is a δgα − nbd. M of b whereas M B = . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 744 https://internationalpubls.com Proof: Assume that B is a δgα − C, then Bc is δgα − O. By definition Bc is δgα − nbd. of each of its points. Let us assume that bc then there is a δgα − O M whereas bM  Bc. So M  B =. Theorem 4.6. If x is an element in the TS (X, ) then 1. δgα − N(x) is non empty. 2. If a sub B δgα − N(x) then xB. Proof: (1) Since X  δgα − N(x) and δgα-N(x) is not empty. (2) Assume that B δgα − N(x), then there is a δgα − O M whereas xM  . Hence x. Theorem 4.7. If a sub B δgα − N(x) and B  , then A δgα − N(x). Proof: Assume that B δgα − N(x), then there is a δgα − O U whereas xU  B. Given B  A, then xU  . Hence A δgα − N(x). Theorem 4.8. Let (X, ) be a TS. If N is a nbd. of t  X, then N is a δgα − nbd. of X. Proof: Assume that N is a nbd. of t  X. By definition there exist an O H whereas t F  N. But we know that all O s are δgα − O whereas t F  N. Thus, N is δgα − nbd. of X. 5. δgα-Derived In this paper we establish the notion of δgα − derived in TSs. Definition 5.1: If M is a sub of a TS (X, ), then a pt. p X is called an δgα − limit point of a M X if every δgα-O S X containing p, contains a pt. of M other than p. The set of all δgα − limit pt. of M is called an δgα-derived set of M and is signified by δgα-D(M). Theorem 5.2: The following five results are true. If M and S are two subs of a TS (X, ). (i). If M  S, then δgα-d(M)  δgα-d(S). (ii) M is an δgα-C if and only if it contains each of its δgα-limit point. (iii). δgα-cl(M) = M ∪ δgα-d(M). (iv). δgα-d(M∪S) ⊇ δgα-d(M) ∪ δgα-d(S). (v). δgα-d(M⋂S) ⊆ δgα-d(M) ⋂ δgα-d(S). Proof: (i) By definition 5.1, we have p δgα-d(M) if and only if E ⋂ (M-{p})≠ φ, for every δgα-O E containing p. But, M⊆S, then E ⋂ (S-{p})≠ φ, for every δgα-O E containing p. Hence pδgα- d(S). Therefore, δgα-d(M)  δgα-d(S). (ii). Let M be an δgα-C and pM then p  (X-M) which is an δgα-O , hence there exist an δgα-O (X- M) whereas (X-M) ⋂ M = φ. So p  δgα-d(M), therefore, δgα-d(M) ⊆ M. Conversely, suppose that δgα-d(M) ⊆ M and pM. Then pδgα-d(M), hence there exist Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 745 https://internationalpubls.com an δgα-O E containing p whereas E ⋂ M = φ and hence X-M = Up     is δgα-O}. Therefore, M is δgα- C. (iii). Since δgα-d(M) ⊆ δgα-cl(M) and M ⊆ δgα-cl(M). δgα-d(M) ∪ M ⊆ δgα-cl(M). Conversely, suppose that p δgα-d(M) ∪ M. Then p δgα-d(M), p M and hence there exist an δgα-O E containing p whereas E⋂M = φ. Thus p  δgα-cl(M)). δgα-cl(M) ⊆ δgα-d(M) ∪ M, therefore, δgα-cl(M) = δgα-d(M) ∪M. (iv). Since M ⊆ M ∪ S and S ⊆ M ∪ S. We have, δgα-d(M) ⊆ δgα-d(M∪S) and δgα-d(S) ⊆ δgα- d(M∪S). Therefore, δgα-d(M) ∪ δgα-d(S) ⊆ δgα-d(M∪S). (v). Since M ⊇ M ⋂ S and S ⊇ M⋂S. We have, δgα-d(M) ⊇ δgα-d(M⋂S) and δgα-d(S) ⊇ δgα- d(M⋂S). Therefore, δgα-d(M) ⋂ δgα-d(S) ⊇ δgα-d(M⋂S). 6. Conclusion In this study, different idea of closure and interior sets namely, δgα-closure, δgα-interior was established and also discussed about δgα-nbd, δgα-derived sets and also about their properties in topological spaces. References [1] Sivaraj, D., & Sasikala, V. E. (2016). A study on soft α–O sets. IOSR Journal of Mathematics, 12(5), 70-74. [2] Kavitha, V., & Sasikala, V. E. (2022). Beta generalized closed sets in topological spaces. Journal of Algebraic Statistics, 13(3), 891-898. 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