Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2662 https://internationalpubls.com Decomposition of (Ψαhb, Δ)-Continuity R. Ramesh ∗, T. Muthukumar, K. Kalaiselvi and L. Senthil Kumar Department of Mathematics, Dr. Mahalingam College of Engineering and Technology, Pollachi, Tamil Nadu, India. ∗E-mail: rameshwaran141@gmail.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: In this article, we explore and introduce the variety of open sets in H-GTS. Additionally, we get the decomposition of (ΨαHb, δ)-C. Keywords: hereditary generalized topology, α-Hb-open, π-Hb-open and π-Hb- open sets. 1. Introduction Let Z be a nonempty set and ϑ be a collection from the subsets of Z. Then ϑ is called a generalized topology (briefly GT) [1], iff ∅∈ϑ and union of open sets of ϑ is open in ϑ. The closure of a subset B of Z, denoted by cϑ(B) is the smallest ϑ- closed sets containing B and the interior (resp. ϑ-σ-interior) of B, denoted by iϑ(B) (resp. iσ(B)) is the largest ϑ-open (resp. ϑ-σ-open) sets contained in B. Definition 1.1. A subset B⊂Z is called 1. ϑ-α-open [2], if B⊂iϑcϑiϑ(B). 2. ϑ-σ-open [2], if B⊂cϑiϑ(B). 3. ϑ-π-open [2], if B⊂iϑcϑ(B). 4. ϑ-β-open [2], if B⊂cϑiϑcϑ(B). 5. ϑ-b-open [14], if B⊂cϑiϑ(B)∪iϑcϑ(B). Definition 1.2. A collection H of subsets of Z is called as a hereditary class [3], if B∈H and V⊂B, then V∈H. Definition 1.3. For a hereditary class H on Band B⊂Z, we define B∗(H, ϑ) = {a ∈ Z: B ∩ V ∈/ H for all V∈ϑ such that a ∈ V} [3]. Definition 1.4. A subset B ⊂ Z is called 1. α- H- o p en [3], if B ⊆ iϑcϑ ∗ iϑ(B), 2. σ-H-o pen [3], if B ⊆ cϑ ∗ iϑ(B), 3. π-H- o p en [3], if B ⊆ iϑcϑ ∗ (B), 4. β-H- o p en [3], if B ⊆ cϑiϑcϑ ∗ (B), 5. ϑ∗ - c losed [3], if cϑ ∗ (B) ⊂ B. 6. b-H- o p e n [8], if B⊆ iϑcϑ ∗ (B) ∪ cϑ ∗ iϑ(B) 7. σϑ∗ - c losed [5], if Bσ ∗⊆B Definition 1.5. A subset B ⊂ Z is said to be 1. α-Hσ-o p en [11], if B⊆iϑcσ ∗ iϑ(B), 2. σ-Hσ-o p e n [11], if B⊆cσ ∗ iϑ(B), 3. π-Hσ- o p e n [11], if B⊆iϑcσ ∗ (B), 4. β-Hσ-o p en [11], if B⊆cϑiϑcσ ∗ (B). 5. b-Hσ-o p e n [10], if B⊆iϑcσ ∗ (B)∪cσ ∗ iϑ(B). Definition 1.6. Consider B be a subset of H-GTS (Z, ϑ, H). Then Bb ∗(H, ϑ)={z∈Z: B∩V∈/H for All V∈ϑ-b-open such that z∈V}. Consider (Z, ϑ, H) be a hereditary generalized topological space. For B⊂Z, define cb ∗(B)=B ∪Bb ∗(H, ϑ) and cb ∗(B) is enlarging, monotone and idempotent. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2663 https://internationalpubls.com Definition 1.7. [12] A subset B ⊂ Z is called 1. ∆- Hψ- o pen, if B⊆iσcσ ∗ iσ(B). 2. Σ-Hψ-o p en , if B⊆cσ ∗ iσ(B). 3. Φ-Hψ-o p en , if B⊆iσcσ ∗ (B). 4. Ω-Hψ-o p en , if B⊆cϑiσcσ ∗ (B). 5. B-Hψ-o p e n , if B⊆cσ ∗ iσ∪iσcσ ∗ (B). Definition 1.8. [13] A subset B of a H - GT S (Z, ϑ, H) is called as 1. α-Hb-open, if B⊆iϑcb∗iϑ(B) 2. σ-Hb-open, if B⊆cb∗iϑ(B) 3. π-Hb-open, if B⊆iϑcb∗(B) 4. β-Hb-open, if B⊆cϑiϑcb∗(B) 5. b-Hb-open, if A⊆iϑcb∗(A)∪cb∗iϑ(A). 2. Generalized ΨHb -open sets Definition 2.1. A subset B of a H - GT S (Z, ϑ, H) is called as 1. Ψα-Hb -open, if B ⊆ iσcb ∗iσ(B) 2. Ψσ-Hb -open, if B ⊆ cb ∗iσ(B) 3. Ψπ-Hb -open, if B ⊆ iσcb ∗(B) 4. Ψβ-Hb -open, if B ⊆ cϑiσcb ∗(B) 5. Ψb-Hb -open, if B ⊆ iσcb ∗(B) ∪ cb ∗iσ(B). The sets Ψα-Hb-open (resp. Ψσ-Hb-open, Ψπ-Hb-open, Ψβ-Hb-open, Ψb-Hb-open, ϑ-pen) denoted by Ψα-HbO(Z) (resp. Ψσ-HbO(Z), Ψπ-HbO(Z), Ψβ-HbO(Z), Ψb-HbO(Z), ϑO(Z)). Theorem 2.2. In H - GT S (Z, ϑ, H) : 1. Any ϑO(Z) set is Ψα-HbO(Z). 2. Any ϑO(Z) set is Ψσ-HbO(Z). 3. Any ϑO(Z) set is Ψπ-HbO(Z). 4. Any ϑO(Z) set is Ψβ-HbO(Z). 5. Any ϑO(Z) set is Ψb-HbO(Z). Proof. (1). Consider a subset B of H -GT S (Z, ϑ, H) is ϑO(Z). Then B⊆iϑ(B)⊆iϑ ∗(B)⊆iϑcb ∗iϑ(B)⊆iσcb ∗iσ(B). Hence B is Ψα-HbO(Z). (2). Consider a subset B of H-GTS (Z, ϑ, H) is ϑO(Z). Then, B⊆iϑ(B)⊆cb ∗iϑ(B)⊆cb ∗iσ(B). Hence B is Ψσ-HbO(Z). (3). Consider a subset B of H-GTS (Z, ϑ, H) is ϑO(Z). Then, B⊆iϑ(B)⊆iϑ cb ∗(B)⊆ iσ cb ∗(B). Hence B is Ψπ-HbO(Z). (4). Consider a subset B of H-GTS (Z, ϑ, H) is ϑO(Z). Then, B⊆iϑ(B)⊆cϑiϑ cb ∗(B)⊆cϑiσcb ∗(B). Hence B is Ψβ-HbO(Z). (5). Consider a subset B of H-GTS (Z, ϑ, H) is ϑO(Z). Then, B⊆iϑ(B)⊆iϑc∗(B)⊆iϑ cb ∗(B)∪cb ∗iϑ(B)⊆iσcb ∗(B)∪cb ∗iσ(B). Hence, B is Ψb-HbO(Z). Theorem 2.3. In H - GT S (Z, ϑ, H): 1. Any α-HbO(Z) is Ψα-HbO(Z). 2. Any σ-HbO(Z) is Ψσ-HbO(Z). 3. Any π-HbO(Z) is Ψπ-HbO(Z). 4. Any β-HbO(Z) is Ψβ-HbO(Z). 5. Any b-HbO(Z) is Ψb-HbO(Z). Proof. 1. Consider B be α-HbO(Z). Then we have, B⊆iϑcb ∗iϑ(B)⊆iσcb ∗iσ(B). Hence Ψα-HbO(Z). 2. Consider B be σ-HbO(Z). Then we have, B⊆cb ∗iϑ(B)⊆ cb ∗iσ(B). Hence Ψσ-HbO(Z). 3. Consider B be π-HbO(Z). Then we have, B⊆iϑcb ∗(B) ⊆ iσcb ∗ (B). Hence Ψπ-HbO(Z). 4. Consider B be β-HbO(Z). Then we have, B⊆ cϑiϑcb ∗(B) ⊆ cϑiσcb ∗ (B). Hence Ψβ-HbO(Z). 5. Consider B be b-HbO(Z). Then we have, B ⊆ iϑcb ∗(B) ∪ cb ∗iϑ(B) ⊆ iσcb ∗(B) ∪ cb ∗iσ(B). Hence Ψb-HbO(Z). Theorem 2.4. In H - GT S (Z, ϑ, H): 1. Any Ψα-HbO(Z) is ∆-HO(Z). 2. Any Ψσ-HbO(Z) is Σ-HO(Z). 3. Any Ψπ-HbO(Z) is Φ-HO(Z). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2664 https://internationalpubls.com 4. Any Ψβ-HbO(Z) is Ω - bO(Z). 5. Any Ψb - HbO(Z) is B -HbO(Z). Proof. 1. Consider B be Ψα-HbO(Z). Then we have, B⊆iσcb ∗iσ(B) ⊆ iσcσ ∗ iσ(B). Hence ∆-HO(Z). 2. Consider B be Ψσ-HbO(Z). Then we have, B⊆cb ∗iσ(B)⊆cσ ∗ iσ(B). Hence Σ-HO(Z). 3. Consider B be Ψπ-HbO(Z). Then we have, B⊆iσcb ∗(B)⊆iσcσ ∗ (B). Hence B is Φ-HO(Z). 4. Consider B be Ψβ-HbO(Z). Then we have, B⊆cσiσcb ∗(B)⊆cσiσcσ ∗ (B). Hence B is Ω - HbO(Z). 5. Consider B be Ψb - HbO(Z). Then we have, B ⊆ iσc∗(B) ∪ cb ∗iσ(B)⊆ iσcσ ∗ (B) ∪ cσ ∗ iσ(B). Hence B is B - HbO(Z). Theorem 2.5. If B is Ψα - HbO(Z), then it is α(σ) -open. Proof. Consider a subset B of H-GTS (Z, ϑ, H) is Ψα-HbO(Z). Then, B⊆iσcb ∗iσ(B)⊆iσcσ ∗ iσ(B) ⊆ iσcσiσ(B). Hence B is α(σ) -open. Theorem 2.6. If B is Ψσ-HbO(Z), then it is σ(σ) -open. Proof. Consider a subset B of H-GT S (Z, ϑ, H) is Ψσ-HbO(Z). Then, B⊆cb ∗iσ(B)⊆cσ ∗ iσ(B) ⊆ cσiσ(B). Hence B is σ(σ) -open. Theorem 2.7. If B is Ψπ-HbO(Z), then it is π(σ) -open. Proof. Consider a subset B of H-GTS (Z, ϑ, H) is Ψπ-HbO(Z). Then, B⊆iσ cb ∗(B) ⊆ iσ cσ ∗ (B) ⊆iσcσ(B). Hence B is π(σ) -open. Theorem 2.8. In H -GT S (Z, ϑ, H), any Ψα-HbO(Z) is Ψσ-HbO(Z). Proof. Consider a subset B of H -GTS (Z, ϑ, H) is Ψα-HbO(Z). Then, B⊆iσ cb ∗iσ(B)⊆ cb ∗iσ(B). Hence Ψσ - HbO(Z). Theorem 2.9. In H-GTS (Z, ϑ, H), any Ψα - HbO(Z) is Ψπ - HbO(Z). Proof. Consider a subset B of H-GTS (Z, ϑ, H) is Ψα-HbO(Z). Then, B⊆iσ cb ∗iσ(B)⊆ iσc∗(B). Hence Ψπ- HbO(Z). Theorem 2.10. A subset B of a H-GT S (Z, ϑ, H), the following results are equivalent. 1. Ψα-HbO(Z) 2. Ψσ- HbO(Z) and Ψπ-HbO(Z). Proof. (1)⇒(2). Consider B is Ψα-HbO(Z). Then by theorem 2.8 and 2.9, Ψσ-HbO(Z) and Ψπ-HbO(Z). (2) ⇒ (1). Consider B is both Ψσ-HbO(Z) and Ψπ-HbO(Z). Then B⊆iσ cb ∗(B) ⊆ iσ cb ∗ cb ∗iσ(B) ⊆ iσ cb ∗iσ(B). Hence Ψα-HbO(Z). Example 2.11. Consider Z={z1, z2, z3, z4, z5}, ϑ={∅, {z1}, {z2}, {z3}, {z1, z2}, {z1, z3}, {z2, z3}, {z1, z2, z3}, {z1, z3, z4}, {z1, z2, z3}, {z1, z2, z3, z4}}, H={∅, {z1}}. Then B = {z1}, {z5}} is Ψσ-HbO(Z) but not Ψα-HbO(Z). Example 2.12. Consider Z={z1, z2, z3, z4}, ϑ={∅, {z1, z3}, {z4}, {z1, z3, z4}, Z} H={∅, {z3}}. Then B={z1, z2, z4} is Ψπ-HbO(Z) but not Ψα-HbO(Z). Remark 2.13. The notions of Ψσ - HbO(Z) and Ψπ-HbO(Z) are independent. Example 2.14. Consider Z = {z1, z2, z3, z4, z5} , ϑ = {∅, {z1}, {z2}, {z3}, {z1, z2}, {z1, z3}, {z2, z3}, {z1, z2, z3}, {z1, z3, z4}, {z2, z3, z4}, {z1, z2, z3, z4}}, H = {∅, {z1}}. Then B = {z1, z3} is Ψσ-HbO(Z) but not Ψπ-HbO(Z). Example 2.15. Consider Z = {z1, z2, z3, z4} ϑ = {∅, {z1, z3}, {z4}, {z1, z3, z4}, Z}, H = {∅, {z3}}. Then B = {z1, z3, z4} is Ψπ-HbO(Z) but not Ψσ-HbO(Z). Theorem 2.16. Any Ψπ-HbO(Z) set is Ψβ-HbO(Z) set. Proof. Consider a subset B of H-GTS (Z, ϑ, H) is Ψπ-HbO(Z). Then B⊆iσcb ∗(B)⊆cϑiσcb ∗(B). Hence B is Ψβ- HbO(Z). Theorem 2.17. Any Ψα-HbO(Z) set is Ψβ-HbO(Z) set. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2665 https://internationalpubls.com Proof. Consider a subset B of H-GTS (Z, ϑ, H) is Ψα-HbO(Z). Then B⊆iσcb ∗iσ(B)⊆iσc∗(B) ⊆cϑiσcb ∗(B). Hence B is Ψβ-HbO(Z). Theorem 2.18. Any Ψσ-HbO(Z) set is Ψβ-HbO(Z) set. Proof. Consider a subset B of H-GTS (Z, ϑ, H) is Ψσ-HbO(Z). Then B⊆cb ∗iσ(B)⊆cϑ ∗ iσ(B)⊆cϑiσ(B)⊆cϑiσcb ∗(B). Hence B is Ψβ-HbO(Z). Theorem 2.19. Any Ψσ - HbO(Z) set is Ψb - HbO(Z) set. Proof. Consider a subset B of H-GTS (Z, ϑ, H) is Ψσ-HbO(Z). Then B⊆cb ∗iσ(B) ⊆ cb ∗iσ(B) ∪ iσcb ∗(B). Hence B is Ψb-HbO(Z). Theorem 2.20. Any Ψπ-HbO(Z) set is Ψb-HbO(Z) set. Proof. Consider a subset B of H-GTS (Z, ϑ, H) is Ψπ-HbO(Z). Then B⊆iσcb ∗(B)⊆cb ∗iσ(B)∪iσcb ∗(B). Hence B is Ψb-HbO(Z). Theorem 2.21. If B⊂Z is both Ψb-HbO(Z) and ϑ-σ-open, then it is Ψβ-HbO(Z). Proof. Let B is both Ψb-HbO(Z) and ϑ-σ-open. Now B⊆iσcb ∗(B) ∪ cb ∗iσ(B)⊆cb ∗(B) and cϑiϑ(B) ⊆ cϑiσ(B) ⊆ cϑiσcb ∗(B). Hence B is Ψβ-HbO(Z). Theorem 2.22. If B⊂Z is both Ψb-HbO(Z) and ϑ-σ-open, then it is ϑ-β-open. Proof. Let B is both Ψb-HbO(Z) and ϑ-σ-open. Then B⊆iσcb ∗(B) ∪ cb ∗iσ(B) and B⊆cϑiϑ(B). Now B⊆iϑcb ∗(B) ∪cb ∗iϑ(B)⊆cb ∗(B), which implies cϑiϑ(B)⊆cϑiϑcb ∗(B)⊆cϑiϑcb ∗ (B)⊆cϑiϑcϑ(B) So B⊆cϑiϑ(B)⊆cϑiϑcϑ(B). Hence B is ϑ - β -open. Theorem 2.23. If B ⊂ Z is both Ψb-HbO(Z) and ϑ∗ -closed, then it is Ψσ-HbO(Z). Proof. Let B is both Ψb-HbO(Z) and ϑ∗-closed. Then B⊆iσcb ∗(B) ∪ cb ∗iσ(B) and c∗(B)⊆B. Now B⊆iσcb ∗(B)∪ cb ∗iσ(B)⊆cb ∗iσ(B)∪iσ(B)=cb ∗iσ(B). Hence B is Ψσ-HbO(Z). Theorem 2.24. If B⊂Z is both Ψb-HbO(Z) and ϑ∗-closed, then it is σ(σ)-open. Proof. Let B is both Ψb-HbO(Z) and ϑ∗-closed. Then B⊆iσcb ∗(B) ∪ cb ∗iσ(B) and cb ∗(B)⊆B. Now B⊆iσcb ∗(B)∪cb ∗iσ(B)⊆cb ∗iσ(B)∪iσ(B)⊆cb ∗iσ(B)⊆cb ∗ iσ(B)⊆cσiσ(B). Hence B is σ(σ)-open. Theorem 2.25. If B⊂Z is both Ψb-HbO(Z) and bϑ∗-closed, then it is Ψσ-HbO(Z). Proof. Let B is both Ψb-HbO(Z) and bϑ∗-closed. Then B⊆iσcb ∗(B) ∪ cb ∗iσ(B) and cb ∗(B) ⊆ B. Now B ⊆ iσcb ∗(B) ∪ cb ∗iσ(B) ⊆ cb ∗iσ(B) ∪ iσ(B) = cb ∗iσ(B). Hence B is Ψσ-HbO(Z). Theorem 2.26. If B⊂Z is both Ψb-HbO(Z) and bϑ∗-closed, then it is σ(σ)-open. Proof. Let B is both Ψb-HbO(Z) and bϑ∗-closed. Then B⊆iσcb ∗(B) ∪ cb ∗iσ(B) and cb ∗(B)⊆B. Now B⊆iσcb ∗(B)∪cb ∗iσ(B)⊆cb ∗iσ(B)∪iσ(B)⊆cb ∗iσ(B)⊆cb ∗ iσ(B)⊆cσiσ(B). Hence B is σ(σ)-open. Theorem 2.27. If B⊂Z is Ψb-HbO(Z) such that iσ(B)=∅, then it is Ψπ-HbO(Z). Proof. Let B be a Ψb-HbO(Z) and iσ(B)=∅. Then B⊆iσcb ∗(B)∪cb ∗iσ(B)=iσcb ∗(B). Hence B is Ψπ-HbO(Z). Theorem 2.28. If B⊂Z is both Ψπ-HbO(Z) and ϑ∗ -closed, then it is σ(σ) - open. Proof. Let B is both Ψπ-HbO(Z) and ϑ∗-closed. Then B⊆iσcb ∗(B) and cb ∗(B)⊆B. Now B⊆iσcb ∗(B)⊆iσ(B). Hence B is ϑ-σ-open. Theorem 2.29. If B⊂Z is both Ψπ-HbO(Z) and bϑ∗-closed, then it is σ(σ) - open. Proof. Let B is both Ψπ-HbO(Z) and bϑ∗-closed. Then B⊆iσcb ∗(B) and cb ∗(B)⊆B. Now B⊆ iσcb ∗(B)⊆iσ(B). Hence B is ϑ-σ-open. 3. Decomposition of (ΨαHb, δ) -Continuity Definition 3.1. A map j:(Z, ϑ, H)→ (W, δ) is (ΨαHb, δ) -continuous ((ΨαHb, ϑ)-C ), if j−1(V) is Ψα-HbO(Z) for each δO(Z) set V in (W, δ). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2666 https://internationalpubls.com Definition 3.2. A map j: (Z, ϑ, H)→ (W, δ) is (ΨσHb, δ) -continuous ((ΨσHb, ϑ) -C ), if j−1(V) is Ψσ-HbO(Z) for each δO(Z) set V in (W, δ). Definition 3.3. A map j: (Z, ϑ, H)→ (W, δ) is (ΨπHb, δ)-continuous ((ΨπHb, ϑ) -C ), if j−1(V) is Ψπ-HbO(Z) for each δO(Z) set V in (W, δ). Theorem 3.4. For a map j : (Z, ϑ, H) → (W, δ , the following results are equivalent. 1. j is (ΨαHb, ϑ) - C. 2. j is (ΨσHb, ϑ) - C and (ΨπHb, ϑ) - C. Proof. Proof is trivial from Theorem 2.14. 4. Conclusion In this paper, we introduced Ψα - HbO(Z), Ψσ - HbO(Z), Ψπ - HbO(Z), Ψβ - HbO(Z), and Ψb - HbO(Z) sets and obtained decomposition of (ΨαHb, ϑ) - C. In future work we will introduce new types of generalized open sets related to these sets and obtain new decomposition of (ϑ, δ) - C. References 1. Csaszar, Generalized topology generalized continuity Acta Mathematica Hungarica 96 (2002), 351-357. 2. Csaszar, Generalized open sets in generalized topologies Acta Mathematica Hungarica 106 (2005), 53-56. 3. 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