Theory of g-Tg-Sets in a Tg-Space

Theory of g-Tg-Sets in a Tg-Space Mohammad Irshad KHODABOCUS and Noor-Ul-Hacq SOOKIA Department of Mathematics, University of Mauritius

Abstract

Results: Table

Several specific types of g-T-sets of a T -space and g-Tg-sets of a Tg-space have been defined in terms of cl, int : P (Ω) → P (Ω) in the literature of T , Tg-spaces. When cl (·), int (·) 7→ clg (·), intg (·) and then combined among themselves, the study of the new type of g-Tg-sets so obtained becomes highly interesting. In this poster, fundamental properties resulting from the theory of g-Tg-sets are displayed.

[ ] I Theorem [6. Let ] g-S T [ g ]be a given g-class in a Tg-space Tg. Then Tg-S : g-S Tg → g-S Tg forms a g-topology on Ω in the Tg-space. I Theorem 7. Let Tg,1 (Ω), Tg,2 (Ω), . . ., Tg,n (Ω) be n ≥ 1

×ν∈I Tg,ν[ (Ω) ]be the Tg-space product. If (Sg,1, Sg,2, . . . , Sg,n ) ∈ ×ν∈I g-S Tg,ν , then [ ] ×ν∈I Sg,ν ∈ g-S ×ν∈I Tg,ν (Ω) . def

Tg-spaces and Tg (Ω) =

∗ n

∗ n

∗ n

∗ n

Hypothesis and Objective I Hypothesis: ∅ ⊆ T ⊆ g-T and ∅ ⊆ Tg ⊆ g-Tg. I Objective: To prove that ∅ ⊆ T ⊆ g-T and ∅ ⊆ Tg ⊆ g-Tg are true.

Results: Diagrams

Introduction Several classes of g-T, g-Tg-sets have been presented by mathematicians in the literature of T , Tg-spaces [1, 2, 3]. By expressing the resulting two operators obtained after taking the ”closure operator clg (·) and its complement” and the ”interior operator clg (·) and its complement,” by means of a so-called g-operator and its complementary g-operator, respectively, the following lines will show how further contributions can be added to the field in a unified way.

Figure 1: Relationships: Classes of T, g-T, Tg, g-Tg-open sets.

Mathematical Framework I Tg-Space: The one-valued map Tg : (P (Ω) → P)(Ω) satisfying ∪ ∪ Tg (∅) = ∅, Tg (Og) ⊆ Og, and Tg ν∈I ∗ Og,ν = ν∈I ∗ Tg (Og,ν ) def

∞

∞

is called a ”g-topology on Ω,” and Tg = (Ω, Tg) is called a ”Tg-space.” I g-Operation: The mapping opg : P (Ω) → P (Ω) is called a ”g-operation” on P (Ω) if the following statements hold: { } { } { } ∀Sg ∈ P (Ω) \ ∅ , ∃ (Og, Kg) ∈ Tg \ ∅ × ¬Tg \ ∅ : ( ) ( ) opg (∅) = ∅ ∨ ¬ opg (∅) = ∅ , ( ) ( ) Sg ⊆ opg (Og) ∨ Sg ⊇ ¬ opg (Kg) , where ¬ opg : P (Ω) → P (Ω) is called the ”complementary g-operation” on P (Ω) and, { } opg ∈ opg,0 (·) , opg,1 (·) , opg,2 (·) , opg,3 (·) { } = intg (·) , clg ◦ intg (·) , intg ◦ clg (·) , clg ◦ intg ◦ clg (·) ; { } ¬ opg ∈ ¬ opg,0 (·) , ¬ opg,1 (·) , ¬ opg,2 (·) , ¬ opg,3 (·) { } = clg (·) , intg ◦ clg (·) , clg ◦ intg (·) , intg ◦ clg ◦ intg (·) . I g-ν-Tg-Set: A g-ν-Tg-sets belongs to [ ] def { ( ) g-ν-S Tg = Sg ⊂ Tg : ∃Og, Kg, opg,ν (·) [( ) ( )]} Sg ⊆ opg,ν (Og) ∨ Sg ⊇ ¬ opg,ν (Kg) . Results: Theorems I

I I

I

I

[ ] Theorem 1. Let opg (·) ∈ Lg Ω be a g-operator in a Tg-space Tg. If Sg,1, Sg,2, . . ., Sg,n ⊂ Tg are n ≥ 1 Tg-sets of the Tg-space Tg, then: (∪ ) ∪ ( ) opg ◦¬ opg ν∈I ∗ Sg,ν = ν∈I ∗ opg ◦¬ opg Sg,ν . (1) n n [ ] Theorem 2. [If S]g,1, Sg,2, . . ., Sg,n ∈ g-S [ ]g-sets ∪ Tg are n ≥ 1 g-T of a class g-S Tg in a Tg-space Tg, then ν∈I ∗ Sg,ν ∈ g-S Tg . [ n] Theorem 3. [If S]g,1, Sg,2, . . ., Sg,n ∈ g-S Tg are n ≥ 1 g-Tg-sets of a class g-S Tg in a Tg-space Tg, then (∩ [ ]) (∩ [ ]) ∨ / g-S Tg . ν∈In∗ Sg,ν ∈ g-S Tg ν∈In∗ Sg,ν ∈ [ ] Theorem 4. Let Sg ∈ g-S Tg in a Tg-space Tg and suppose [( ) ( )] (∃Rg ⊂ Tg) Rg ⊆ opg (Sg) ∨ Rg ⊇ ¬ opg (Sg) [ ] holds, then Rg ∈ g-S Tg . [ ] [ ] [ ] Theorem 5. If g-S Tg = g-O Tg ∪ a[ class [ g-K ] Tg [ denotes ] ] of g-Tg-open and g-Tg-closed sets, and S Tg = O Tg ∪ K Tg denotes and[ Tg]-closed[ sets, [ ]a class of[Tg-open ] ] then[ ] [ ] g-S Tg ⊇ g-O Tg ∪ g-K Tg ⊇ O Tg ∪ K Tg ⊇ S Tg .

Research in Pure Mathematics

Figure 2: Relationships: Classes of T, g-T, Tg, g-Tg-closed sets.

Conclusion I A new theory, called Theory of g-Tg-Sets is presented. The very first advantage is that the theory holds equally well when (Ω, Tg) = (Ω, T ) and other features adapted on this basis. Hence, in a Tg-space the theoretical framework categorises such pairs of concepts as g-Tg-sets, g-Tg-semi-sets, g-Tg-pre-sets, and g-Tg-semi-pre-sets as g-Tg-sets of categories 0, 1, 2, and 3, respectively, and theorises the concepts in a unified way. I It is an interesting topic for future research to develop the theory of g-Tg-sets of mixed categories: For some pair (ν, µ) ∈ I30 × I30 such that ν { ̸= µ, to develop the theory of g-Tg-sets belonging [ ]to [ ]} {Og = Og,ν ∪ Og,µ : (Og,ν , Og,µ) ∈ g-ν-O[ Tg] × g-µ-O[ Tg]} , Kg = Kg,ν ∪ Kg,µ : (Kg,ν , Kg,µ) ∈ g-ν-K Tg × g-µ-K Tg . References ´ Cs´asz´ar. [1] A. Generalized open sets. Acta Math. Hungar., 75((1-2)):65–87, 1997. [2] A. Gupta and R. D. Sarma. A note on some generalized closure and interior operations in a topological space. Math. Appl., 6:11–20, 2017. [3] B. K. Tyagi and H. V. S. Chauhan. On generalized closed sets in generalized topological spaces. CUBO A Mathematical Journal, 18(51):27–45, 2016.

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