Theory of Generalized Sets in Generalized Topological Spaces

New Theory

ISSN: 2149-1402

Journal of New Theory https://dergipark.org.tr/en/pub/jnt

Open Access

Editor-in-Chief Naim Çağman

www.dergipark.org.tr/en/pub/jnt

Theory of Generalized Sets in Generalized Topological Spaces

Mohammad Irshad Khodabocus^{1 ID} , Noor-Ul-Hacq Sookia^{2 ID}

Article History Received : 13 Mar 2021 Accepted : 20 Sep 2021 Published : 30 Sep 2021

10.53570/jnt.896345 Research Article

Abstract− Several specific types of generalized sets (briefly, g-Tg-sets) in gener- alized topological spaces (briefly, Tg-spaces) have been defined and investigated for various purposes from time to time in the literature of Tg-spaces. Our recent re- search in the field of a new class of g-Tg-sets in Tg-spaces is reported herein as a starting point for more generalized classes. It is shown that the class of g-Tg-sets is a superclass of those whose elements are called open, closed, semi-open, semi-closed, pre-open, pre-closed, semi-pre-open, and semi-pre-closed sets in a Tg-space. A sub- class of the Tg-subspace corresponds to the class of g-Tg-sets of a Tg-space. A class of g-Tg-sets of the Cartesian product of these Tg-spaces corresponds to the Cartesian product of a finite number of classes of g-Tg-sets, each of which belongs to a Tg-space.

Diagrams establish the various relationships amongst the classes presented here and in the literature, and an ad hoc application supports the overall theory.

Keywords − Generalized topology, generalized topological space, generalized operations, generalized open sets, generalized closed sets

Mathematics Subject Classification (2020) − 54A05, 54B05

1. Introduction

Just as the notion of T -set (open or closed set relative to ordinary topology) is fundamental and indispensable in the study of T-sets in T -spaces (arbitrary sets in ordinary topological spaces) and in the formulation of the concept of g-T -set (generalized T -open or T -closed set relative to ordinary topology) in the study of g-T-sets in T -spaces (generalized sets in ordinary topological spaces) [1–6], so is the notion of Tg-set (open or closed set relative to generalized topology) in the study of Tg-sets in Tg-spaces (arbitrary sets in generalized topological spaces) and in the formulation of the concept of g-T_g-set (generalized T_g-open or T_g-closed set relative to generalized topology) in the study of g-T_g- sets in Tg-spaces (generalized sets in generalized topological spaces) [7]. Thus, the g-topology maps T_g : P (Ω) −→ P (Ω) from the power set P (Ω) of Ω into itself, thereby inducing g-topologies on the underlying set Ω, are classes of distinguished open subsets of a T -space which are not T -open sets but are T_g-open sets which are related to the families of g-T -open sets [8, 9]. Examples of g-T-sets in T -spaces are α-open and α-closed sets [10], β-open sets [11], and γ-open sets [12]. Examples of g-T_g-sets in T_g-spaces are ∆_µ-sets and ∇_µ-sets [13], ω-open sets [2], and θ-sets [14]. From these α, β, γ-sets and, ∆_µ, ∇_µ, ω, θ-sets, the theories of g-T-sets and g-T_g-sets then appear to be subjects of primary interest.

To the best of our knowledge, the theory of g-T-sets is well-known and that of g-T_g-sets less- known. The earliest works on the theory of g-T-sets are those of Levine [15, 16], Nj˚astad [10], and

1ikhodabo@gmail.com (Corresponding Author);²sookian@uom.ac.mu

1Department of Emerging Technologies, Faculty of Sustainable Development and Engineering, Universit´e des Mas- careignes, Rose Hill Campus, Mauritius

2,2Department of Mathematics, Faculty of Science, University of Mauritius, R´eduit, Mauritius

Cs´asz´ar [14, 17–20], and the latest works on the theory of g-T-sets are those of Rajeshwari et al. [21], Jeyanthi et al. [3, 13], Ghour et al. [2], and Tyagi et al. [6], among others. Levine [16] introduced and investigated the weaker forms of open sets, Nj˚astad [10] introduced and investigated the structures of some classes of more or less nearly open sets, and Cs´asz´ar [20] introduced the notion of g-topologies; [21]

introduced the weaker forms of closed sets and studied some of their characterizations, Jeyanthi et al. [3] gave a unified framework for the study of several types of g-T_g-sets, Ghour et al. [2] extended the notion of a type of g-T-sets in a T -space to its analogue in a T_g-space, and Tyagi et al. [6] introduced and investigated several types of g-Tg-sets in Tg-spaces.

Several other specific classes of g-T, g-T_g-sets have been defined and investigated by other authors for various purposes from time to time in the literature of T , T_g-spaces [9, 22–38]. The fruitfulness of all these references have made significant contributions to the theory of T , Tg-spaces, among others.

In this paper, we will show how further contributions can be added to the field in a unified way.

The rest of this paper is structured in this manner: In Section 2, preliminary notions are described in Subsection 2.1 and the main results of the theory of g-Tg-sets in Tg-spaces are reported in Section 3. In Section 4, the establishment of the various relationships between the classes of Tg-open and T_g-closed sets and the classes of g-T_g-open and g-T_g-closed sets in the T_g-space T_g are discussed and illustrated through diagrams in Subsection 4.1. To support the work, a nice application, concentrating on fundamental concepts from the standpoint of the theory of g-Tg-sets is presented in Subsection 4.2.

Finally, Subsection 5 provides concluding remarks and future directions of the theory of g-T_g-sets in T_g-spaces.

2. Theory

2.1. Preliminaries

Our discussion starts by recalling a carefully chosen set of terms used in this study [39]. Throughout this manuscript, the structures T = (Ω, T ) and T_g = (Ω, T_g), respectively, are called ordinary and generalized topological spaces (briefly, T -space and T_g-space). The symbols T and T_g, respectively, are called ordinary topology and generalized topology (briefly, topology and g-topology). Subsets of T and Tg, respectively, are called T-sets and Tg-sets; subsets of T and Tg, respectively, are called T -open and T_g-open sets, and their complements are called T -closed and T_g-closed sets. Generalizations of T-sets, T -open and T -closed sets in T , respectively, are called g-T-sets, g-T -open and g-T -closed sets;

generalizations of Tg-sets, Tg-open and Tg-closed sets in Tg, respectively, are called g-Tg-sets, g-Tg-open and g-T_g-closed sets; U stands for the universe of discourse, fixed within the framework of the theory of g-Tg-sets and containing as elements all sets (Ω, Γ-sets; T , g-T , T, g-T-sets; Tg, g-Tg, Tg, g-Tg-sets) considered in this theory, and I_n⁰ := ν ∈ N⁰ : ν ≤ n ; index sets I_∞⁰ , I_n^∗, I_∞^∗ are defined similarly.

A set Γ ⊂ U is a subset of the set Ω ⊂ U and, for some T_g-open set O_g ∈ T ∪ g-T ∪T_g∪ g-T_g, these implications hold:

O_g∈ T ⇒ O_g∈ g-T ⇒ O_g∈ T_g ⇒ O_g∈ g-T_g ⇒ O_g ⊂ Ω ⊂ U (1) In a natural way, a monotonic map T_g : P (Ω) −→ P (Ω) from the power set P (Ω) of Ω into itself can be associated to a given mapping π_g: Ω −→ Ω, thereby inducing a g-topology T_g⊂ P (Ω) on the underlying set Ω [9]. Therefore, the definition of a Tg-space can be presented in a nice way. Thus, retaining the axioms to be satisfied by its g-topology [33], and assuming no separation axioms, unless otherwise stated, the following definition is suggestive:

Definition 2.1 (T_g-Space [39]). Let Ω ⊂ U be a given set and let P (Ω) :=O_g,ν : O_g,ν ⊆ Ω be the family of all subsets O_g,1, O_g,2, . . ., of Ω. Then, every one-valued map of the type T_g: P (Ω) −→ P (Ω) satisfying the following axioms:

i. Tg(∅) = ∅ ii. T_g(O_g) ⊆ O_g

iii. T_g S

ν∈I_∞^∗ O_g,ν
= S_ν∈I∗

∞T_g(O_g,ν)

is called a “g-topology on Ω,” and the structure Tg:= (Ω, Tg) is called a “Tg-space.”

In Definition 2.1, by Ax. i., Ax. ii., and Ax. iii., respectively, are meant that the unary operation T_g: P (Ω) −→ P (Ω) preserves nullary union, is contracting and preserves binary union. Any element O_g∈ T_g(Ω) of the T_g-space T_gis called a T_g-open set and its complement element ∁ (Og) = K_g∈ T/ _g(Ω) is called a Tg-closed set. If there exists a ν ∈ I_∞^∗ such that Og,ν = Ω, then Tg is called a strong Tg- space [9,19]. Moreover, if the relation T_g T

ν∈I_n^∗O_g,ν
= T_ν∈I∗

nT_g(Og,ν) holds for any index set I_n^∗ ⊂ I_∞^∗ such that n < ∞, then T_g is called a quasi T_g-space [17].

Definition 2.2 (g-Closure, g-Interior Operators [39]). Let T_g be a T_g-space on the set Ω ⊂ U with a g-topology Tg: P (Ω) −→ P (Ω). Then,

i. The operator clg : P (Ω) −→ P (Ω) carrying each Tg-set Sg ⊂ T_g into its closure clg(Sg) = T_g\ int_g(T_g\ S_g) ⊂ T_g is called a “g-closure operator.”

ii. The operator int_g : P (Ω) −→ P (Ω) carrying each T_g-set S_g ⊂ T_g into its interior int_g(S_g) = T_g\ cl_g(Tg\ S_g) ⊂ Tg is called a “g-interior operator.”

By convention, we let Tg(Ω) and ¬Tg(Ω), respectively, stand for the classes of all Tg-open and T_g-closed sets relative to the g-topology T_g. Their proper definitions are contained below.

Definition 2.3 (Classes: T_g-Open, T_g-Closed Sets [39]). Let T_g be a T_g-space, let ∁ : P (Ω) −→ P (Ω) denotes the absolute complement with respect to the underlying set Ω ⊂ U, and let S_g ⊂ T_g be any T_g-set. The classes

T_g(Ω) :=O_g⊂ T_g : O_g ∈ T_g

and ¬T_g(Ω) :=K_g⊂ T_g: ∁ (Kg) ∈ T_g

(2) respectively, denote the classes of all T_g-open and T_g-closed sets relative to the g-topology T_g, and the classes

C^sub_Tg [Sg] :=O_g∈ T_g: Og⊆ S_g

and C^sup_¬T

g[Sg] :=K_g ∈ ¬T_g: Kg ⊇ S_g

(3) respectively, denote the classes of Tg-open subsets and Tg-closed supersets (complements of the Tg-open subsets) of the T_g-set S_g ⊂ T_g relative to the g-topology T_g.

That C^sub_Tg [S_g] ⊆ T_g(Ω) and ¬T_g(Ω) ⊇ C^sup_¬T

g[S_g] are true for the T_g-set S_g ⊂ T_g in question are clear from the context. To this end, the g-closure and the g-interior of a Tg-set Sg ⊂ T_g in a Tg-space define themselves as

int_g(S_g) := [

Og∈C^sub_Tg[Sg]

O_g and cl_g(S_g) := \

Kg∈C^sup_¬Tg[Sg]

K_g (4)

We note in passing that, clg(·) ̸= cl (·) and intg(·) ̸= int (·), because the resulting sets obtained from the intersection of all T_g-closed supersets and the union of all T_g-open subsets, respectively, relative to the g-topology T_g are not necessarily equal to those which would be obtained from the intersection of all T -closed supersets and the union of all T -open subsets relative to the topology T [23]. Throughout this work, by cl_g◦ int_g(·), int_g◦ cl_g(·), and cl_g◦ int_g◦ cl_g(·), respectively, are meant cl_g(int_g(·)), int_g(cl_g(·)), and cl_g(int_g(cl_g(·))); other composition operators are defined in a similar way. Also, the backslash Tg\ S_g refers to the set-theoretic relative complement of Sg in Tg. Finally, for convenience of notation, let P^∗(Ω) = P (Ω) \∅ , T_g^∗ = T_g\∅ , and ¬T_g^∗= ¬T_g\∅ . Definition 2.4 (g-Operation [39]). Let T_g = (Ω, T_g) be a T_g-space. Then, a mapping op_g: P (Ω) −→

P (Ω) on P (Ω) ranging in P (Ω) is called a “g-operation” if and only if the following statements hold:

∀Sg∈ P^∗(Ω)

∃ (Og, Kg) ∈ Tg^∗× ¬Tg^∗

 op_g(∅) = ∅
∨ ¬ op_g(∅) = ∅
∨ Sg⊆ op_g(Og)
∨ Sg⊇ ¬ op_g(Kg)


(5)

where ¬ op_g : P (Ω) −→ P (Ω) is called the “complementary g-operation” on P (Ω) ranging in P (Ω) and, for all (S_g, U_g,µ, V_g,ν) ∈ N

α∈I₃^∗P^∗(Ω) such that W_g = U_g,µ ∪ V_g,ν and Wˆ_g, ¬ ˆW_g

= op_g(W_g) , ¬ op_g(W_g)
, the following axioms are satisfied:

i. S_g ⊆ op_g(O_g)
∨ S_g⊇ ¬ op_g(K_g)

ii. op_g(S_g) ⊆ op_g◦ op_g(O_g)
∨ ¬ op_g(S_g) ⊇ ¬ op_g◦¬ op_g(K_g)
iii.



Wˆ_g⊆ [

σ=µ,ν

op_g(O_g,σ)

 _

¬ ˆW_g⊇ [

σ=µ,ν

¬ op_g(K_g,σ)



iv. U_g,µ⊆ V_g,ν −→ op_g(Og,µ) ⊆ op_g(Og,ν)
∨ U_g,µ⊇ V_g,ν ←− ¬ op_g(Kg,µ) ⊇ ¬ op_g(Kg,ν)
for some (O_g, O_g,µ, O_g,ν) ∈N

α∈I₃^∗T_g^∗ and (K_g, K_g,µ, K_g,ν) ∈N

α∈I₃^∗¬T_g^∗.

The formulation of Definition 2.5 is based on the axioms of the ˇCech closure operator [25] and the various axioms used by many mathematicians to define closure operators [36]. The class LgΩ stands for the class of all possible g-operators and their complementary g-operators in the T_g-space T_g. Definition 2.5 (op_g(·)-Elements [39]). Let Tg be a Tg-space. The elements of the class LgΩ = L^ω_gΩ × L^κ_gΩ, where

L_gΩ := op_g,νµ(·) = op_g,ν(·) , ¬ op_g,µ(·)
: (ν, µ) ∈ I₃⁰× I₃⁰

(6) in the Tg-space Tg are defined as:

op_g(·) ∈ L^ω_gΩ := op_g,0(·) , op_g,1(·) , op_g,2(·) , op_g,3(·)

= int_g(·) , clg◦ int_g(·) , intg◦ cl_g(·) , clg◦ int_g◦ cl_g(·)

¬ op_g(·) ∈ L^κ_gΩ := ¬ op_g,0(·) , ¬ op_g,1(·) , ¬ op_g,2(·) , ¬ op_g,3(·)

= cl_g(·) , int_g◦ cl_g(·) , cl_g◦ int_g(·) , int_g◦ cl_g◦ int_g(·)

(7) We remark in passing that, op_g,11(·) = ¬ op_g,22(·), and the use of op_g(·) = op_g(·) , ¬ op_g(·)
∈ L_gΩ on a class of T_g-sets will construct a new class of g-T_g-sets, just as the use of LΩ := op_ν(·) = op_ν(·) , ¬ op_ν(·)
: ν ∈ I₃⁰ on the class of T-sets have constructed the new class of g-T-sets. But since cl_g(·) ̸= cl (·) and int_g(·) ̸= int (·), in general, it follows that op_g(·) ̸= op (·) and, therefore, the new class of g-T_g-sets that will be obtained from the first construction will, in general, differ from the new class of g-T-sets that had been obtained from the second construction.

Definition 2.6 (g-ν-T_g-Set [39]). A T_g-set S_g ⊂ T_g in a T_g-space is called a “g-T_g-set” if and only if there exist a pair (Og, Kg) ∈ Tg× ¬T_g of Tg-open and Tg-closed sets, and a g-operator op_g(·) ∈ LgΩ

such that the following statement holds:

(∃ξ)(ξ ∈ S_g) ∧ S_g ⊆ op_g(Og)
∨ S_g ⊇ ¬ op_g(Kg)



(8) The g-Tg-set Sg ⊂ T_g is said to be of category ν if and only if it belongs to the following class of g-ν-Tg-sets:

g-ν-ST_g := S_g⊂ T_g : ∃O_g, Kg, op_g,ν(·)


S_g⊆ op_g,ν(Og)
∨ S_g ⊇ ¬ op_g,ν(Kg)


(9) It is called a g-ν-T_g-open set if it satisfies the first property in g-ν-ST_g and a g-ν-T_g-closed set if it satisfies the second property in g-ν-ST_g. The classes of g-ν-T_g-open and g-ν-Tg-closed sets, respectively, are defined by

g-ν-OT_g

:= S_g⊂ T_g : ∃O_g, op_g,ν(·)
S_g⊆ op_g,ν(Og) g-ν-KT_g

:= S_g⊂ T_g : ∃K_g, op_g,ν(·)
S_g ⊇ ¬ op_g,ν(K_g)

(10) From the class g-ν-ST_g, consisting of the classes g-ν-OT_g and g-ν-KT_g, respectively, of g-ν-T_g- open and g-ν-T_g-closed sets of category ν, where ν ∈ I₃⁰, there results in the following definition.

Definition 2.7 (g-T_g-Set [39]). Let T_g be a T_g-space. If, for each ν ∈ I₃⁰, g-ν-OT_g and g-ν-KT_g, respectively, denote the classes of g-ν-Tg-open and g-ν-Tg-closed sets of category ν then,

g-ST_g = S_ν∈I0

3 g-ν-ST_g

= S

ν∈I₃⁰ g-ν-OT_g ∪ g-ν-KT_g

= S

ν∈I₃⁰g-ν-OT_g
∪ S_ν∈I0

3 g-ν-KT_g

= g-OT_g ∪ g-KT_g

(11) In the sequel, it is interesting to view the concepts of open, semi-open, pre-open, semi-pre-open sets as g-T-open sets of categories 0, 1, 2, and 3; likewise, to view the concepts of closed, semi-closed, pre-closed, semi-pre-closed sets as g-T-closed sets of categories 0, 1, 2, and 3. These can be realised by omitting the subscript “g” in all symbols of the above definitions.

Definition 2.8 (g-ν-T-Set [39]). A T-set S ⊂ T in a T -space is called a “g-T-set” if and only if there exists a pair (O, K) ∈ T × ¬T of T -open and T -closed sets, and an operator op (·) ∈ LΩ such that the following statement holds:

(∃ξ)(ξ ∈ S) ∧ (S ⊆ op (O)) ∨ (S ⊇ ¬ op (K))
 (12) The g-T-set S ⊂ T is said to be of category ν if and only if it belongs to the following class of g-ν-T -sets:

g-ν-ST := S ⊂ T : (∃O, K, op_ν(·))(S ⊆ op_ν(O)) ∨ (S ⊇ ¬ op_ν(K))

(13) It is called a g-ν-T-open set if it satisfies the first property in g-ν-ST and a g-ν-T-closed set if it satisfies the second property in g-ν-ST. The classes of g-ν-T-open and g-ν-T-closed sets, respectively, are defined by

g-ν-OT := S ⊂ T : (∃O, op_ν(·))S ⊆ op_ν(O) g-ν-KT := S ⊂ T : (∃K, op_ν(·))S ⊇ ¬ op_ν(K)

(14) As in the previous definitions, from the class g-ν-ST, consisting of the classes g-ν-OT and g-ν-KT, respectively, of g-ν-T-open and g-ν-T-closed sets of category ν, where ν ∈ I₃⁰, there results in the following definition.

Definition 2.9 (Class: g-Tg-Sets [39]). Let T be a T -space. If, for each ν ∈ I₃⁰, g-ν-OT and g-ν-KT, respectively, denote the classes of g-ν-T-open and g-ν-T-closed sets of category ν then,

g-ST = S_ν∈I0

3g-ν-ST = S_ν∈I0

3 g-ν-OT ∪ g-ν-KT

= S

ν∈I₃⁰g-ν-OT
∪ S_ν∈I0

3 g-ν-KT

= g-OT ∪ g-KT (15)

The classes of T_g-open and T_g-closed sets in a T_g-space T_g as well as the classes of T-open and T-closed sets in a T -space T are defined as thus:

Definition 2.10 (Families: g-T_g-Open Sets, g-T_g-Closed Sets [39]). Let T_g = (Ω, T_g) be a T_g-space and let T = (Ω, T ) be a T -space.

i. The classes O [T_g] and K [T_g] denote the families of T_g-open and T_g-closed sets, respectively, in T_g, with S [Tg] = O [Tg] ∪ K [Tg].

ii. The classes O [T] and K [T] denote the families of T-open and T-closed sets, respectively, in T, with S [T] = O [T] ∪ K [T].

In the following sections, the main results of the theory of g-T_g-sets are presented.

3. Main Results

Theorem 3.1. Let clg : P (Ω) −→ P (Ω) and intg : P (Ω) −→ P (Ω), respectively, be g-closure and g-interior operators in the T_g-space T_g. Then,

i. cl_g(·) and int_g(·) are enhancing and contracting, respectively.

ii. clg(·) and intg(·) are idempotent.

iii. clg(·) and intg(·) are monotone.

Proof.

i. Since the following logical statement

S_g⊂ T_g : (∀ξ)(ξ ∈ cl_g(Sg) ←− ξ ∈ Sg) ∨ (ξ ∈ intg(Sg) −→ ξ ∈ Sg) holds, it follows that Sg⊆ cl_g(Sg) or Sg ⊇ int_g(Sg).

ii. If S_g is open, then S_g = int_g(S_g); if it is closed, S_g = cl_g(S_g). Consequently, the substitutions S_g 7−→ int_g(Sg) and Sg 7−→ cl_g(Sg), respectively, give intg(Sg) = intg◦ int_g(Sg) and clg(Sg) = cl_g◦ cl_g(S_g).

iii. Let Rg, Sg ⊂ T_g such that Rg ⊆ S_g. Then, Rg ⊆ cl_g(Rg), Rg ⊇ int_g(Rg), Sg ⊆ cl_g(Sg), and S_g⊇ int_g(S_g) by i. Consequently, int_g(R_g) ⊆ int_g(S_g) and cl_g(R_g) ⊆ cl_g(S_g).

Lemma 3.2. Let Sg⊂ T_g be a Tg-set of a Tg-space. Then, i. (S_g= ∅) ∧ (Ω ∈ T_g) ⇒ (int_g(S_g) = ∅) ∧ (cl_g(∅) = ∅) ii. (S_g= ∅) ∧ (Ω /∈ T_g) ⇒ (int_g(S_g) = ∅) ∧ (cl_g(∅) ̸= ∅) Proof.

i. If Sg = ∅ and Ω ∈ Tg, then ∅ ∈ C^sub_T

g [∅]
∧ ∅ ∈ C^sup_T

g [∅]
. Consequently, int_g(∅) = ∅ and clg(∅) = ∅.

ii. If Sg = ∅ and Ω /∈ T_g, then ∅ ∈ C^sub_T

g [∅]
∧ ∅ /∈ C^sup_T

g [∅]
. Consequently, int_g(∅) = ∅ and intg(∅) ̸= ∅.

According to Sarsak [40] and Noiri [41], the T_g-space T_gmay be called a µ-space when cl_g(∅) = ∅.

Theorem 3.3. If S_g,1, S_g,2, . . ., S_g,n ⊂ T_g are n ≥ 1 T_g-sets of a T_g-space, then, i. clg S

ν∈I^∗_nS_g,ν
= S_ν∈I∗

nclg S_g,ν
ii. intg T

ν∈I_n^∗S_g,ν
= T_ν∈I∗

nintg S_g,ν

Proof. Expressed in set-builder notation, the g-closure and the g-interior of a Tg-set S_g ⊂ T_g in a T_g-space can also be defined as thus:

cl_g(S_g) := ξ ∈ T_g: (S_g∩ cl (O_g) ̸= ∅) ∧ (ξ ∈ O_g∈ T_g)

intg(Sg) := ξ ∈ T_g: (Sg∩ int (O_g) = int (Og)) ∧ (ξ ∈ Og ∈ T_g)

respectively, from which it is easily seen that, cl_g S

ν∈I_n^∗S_g,ν

= [

ν∈I_n^∗

ξ ∈ T_g: (S_g,ν∩ cl (O_g) ̸= ∅) ∧ (ξ ∈ O_g ∈ T_g)

= ξ ∈ T_g : S

ν∈I_n^∗S_g,ν
∩ cl (O_g) ̸= ∅
∧ (ξ ∈ O_g∈ T_g)

= ξ ∈ T_g : S

ν∈I_n^∗ S_g,ν∩ cl (O_g)
̸= ∅
∧ (ξ ∈ O_g∈ T_g)

= ξ ∈ T_g : W

ν∈I_n^∗ S_g,ν∩ cl (O_g) ̸= ∅
∧ (ξ ∈ O_g∈ T_g)

= S

ν∈I^∗_nclg S_g,ν
Likewise, it is also easily seen that,

int_g T

ν∈I_n^∗S_g,ν

= \

ν∈I_n^∗

ξ ∈ T_g: (Sg,ν∩ int (O_g) = int (O_g)) ∧ (ξ ∈ O_g∈ T_g)

= ξ ∈ T_g: T

ν∈I_n^∗S_g,ν
∩ int (O_g) = int (Og)
∧ (ξ ∈ O_g∈ T_g)

= ξ ∈ T_g: T

ν∈I_n^∗ S_g,ν∩ int (O_g)
= int (O_g)
∧ (ξ ∈ O_g∈ T_g)

= ξ ∈ T_g: V

ν∈I_n^∗ S_g,ν∩ int (O_g) = int (O_g)
∧ (ξ ∈ O_g∈ T_g)

= T

ν∈I_n^∗int_g S_g,ν

Clearly, S_g,µ ⊆S

ν∈I^∗_nS_g,ν and S_g,µ ⊇T

ν∈I_n^∗S_g,ν hold true for any µ ∈ I_n^∗. The following corollary, then, is an immediate consequence of the above theorem.

Corollary 3.4. If S_g,1, S_g,2, . . ., S_g,n ⊂ T_g are n ≥ 1 T_g-sets of a T_g-space, then, i. clg T

ν∈I^∗_nS_g,ν
⊆ T_ν∈I∗

nclg S_g,ν
ii. int_g S

ν∈I_n^∗S_g,ν
⊇ S_ν∈I∗

nint_g S_g,ν

Proposition 3.5. For any Tg-set Sg ⊂ T_g in a Tg-space Tg, the following statement holds:

T_g\ int_g(S_g) ∪ cl_g(T_g\ S_g)
= ∅ (16) Proof. Let ξ ∈ clg(T_g\ S_g). Then, ξ ∈ T_g\ S_g since, T_g\ S_g ⊆ cl_g(T_g\ S_g). But, T_g\ S_g ⊆ T_g\ intg(Sg) ⊆ clg(Tg\ S_g) and, consequently, ξ ∈ Tg\ int_g(Sg). Hence, there follows that, clg(Tg\ S_g) ⊆ T_g\ int_g(Sg). Conversely, let ξ ∈ Tg\ int_g(Sg). Then, ξ ∈ clg(Tg\ int_g(Sg)), since Tg\ int_g(Sg) ⊆ cl_g(T_g\ int_g(S_g)). But, since T_g\ int_g(S_g) ⊆ cl_g(T_g\ S_g) and cl_g(T_g\ S_g) ⊆ cl_g(T_g\ int_g(S_g)), and, consequently, ξ ∈ Tg\ int_g(Sg). Hence, Tg\ int_g(Sg) ⊆ clg(Tg\ S_g). Since clg(Tg\ S_g) = Tg\ int_g(Sg) is equivalent to

cl_g(T_g\ S_g) ⊆ T_g\ int_g(S_g)
∧ cl_g(T_g\ S_g) ⊇ T_g\ int_g(S_g)
the proof of the proposition at once follows.

Proposition 3.6. Let cl_g: P (Ω) −→ P (Ω) and int_g: P (Ω) −→ P (Ω), respectively, be g-closure and g-interior operators in a Tg-space Tg. If Sg,1, Sg,2, . . ., Sg,n⊂ T_g are n ≥ 1 Tg-sets of the Tg-space Tg, then,

i. cl_g◦ int_g S

ν∈I_n^∗S_g,ν
⊇ S_ν∈I∗

ncl_g◦ int_g S_g,ν
ii. intg◦ cl_g S

ν∈I_n^∗S_g,ν
⊇ S_ν∈I∗

nintg◦ cl_g S_g,ν
iii. cl_g◦ int_g◦ cl_g S

ν∈I_n^∗S_g,ν
⊇ S_ν∈I∗

ncl_g◦ int_g◦ cl_g S_g,ν

Proof. Since the relations cl_g S

ν∈I_n^∗S_g,ν
= S_ν∈I∗

ncl_g S_g,ν
, int_g S

ν∈I_n^∗S_g,ν
⊇ S_ν∈I∗

nint_g S_g,ν
hold, it follows that

clg◦ int_g S

ν∈I_n^∗S_g,ν

⊇ cl_g S

ν∈I_n^∗intg S_g,ν

= S

ν∈I_n^∗clg◦ int_g S_g,ν
int_g◦ cl_g S

ν∈I_n^∗S_g,ν

⊇ int_g S

ν∈I^∗_ncl_g S_g,ν

= S

ν∈I_n^∗cl_g◦ int_g S_g,ν
cl_g◦ int_g◦ cl_g S

ν∈I_n^∗S_g,ν

= cl_g◦ int_g S

ν∈I_n^∗cl_g S_g,ν

⊇ S

ν∈I_n^∗clg◦ int_g◦ cl_g S_g,ν

From the above proposition, it is obvious that their duals are int_g◦ cl_g T

ν∈I_n^∗S_g,ν

⊆ T

ν∈I_n^∗int_g◦ cl_g S_g,ν
cl_g◦ int_g T

ν∈I_n^∗S_g,ν

⊆ T

ν∈I_n^∗cl_g◦ int_g S_g,ν
int_g◦ cl_g◦ int_g T

ν∈I_n^∗S_g,ν

⊆ T

ν∈I_n^∗int_g◦ cl_g◦ int_g S_g,ν

(17) respectively. On this basis, we have the following corollary:

Corollary 3.7. Let op_g(·) ∈ LgΩ be a g-operator in a T_g-space Tg. If Sg,1, Sg,2, . . ., Sg,n ⊂ T_g are n ≥ 1 T_g-sets of the T_g-space T_g, then,

i. op_g◦¬ op_g S

ν∈I_n^∗S_g,ν
⊇ S_ν∈I∗

nop_g◦¬ op_g S_g,ν
ii. ¬ op_g◦ op_g T

ν∈I_n^∗S_g,ν
⊆ T_ν∈I∗

n¬ op_g◦ op_g S_g,ν

Theorem 3.8. If S_g,1, S_g,2, . . ., S_g,n ∈ g-ST_g are n ≥ 1 g-T_g-sets of a class g-ST_g in a T_g-space T_g, thenS

ν∈I_n^∗S_g,ν ∈ g-ST_g.

Proof. The statement Sg,ν ∈ g-ST_g for every ν ∈ I_n^∗ is identical to the logical statement:

∃ (O_g,ν, K_g,ν) ∈ T_g× ¬T_g: S_g,ν ⊆ op_g(O_g,ν)
∨ S_g,ν ⊇ ¬ op_g(K_g,ν)
On the other hand, if op_g(·) ∈ LgΩ is a g-operator in the T_g-space, then

op_g S

ν∈I_n^∗O_g,ν

= S

ν∈I_n^∗op_g(Og,ν)

¬ op_g S

ν∈I^∗_nK_g,ν

= S

ν∈I_n^∗¬ op_g(Kg,ν) Consequently,

_

ν∈I_n^∗

S_g,ν ⊆ op_g(Og,ν)
∨ S_g,ν ⊇ ¬ op_g(Kg,ν)

⇒ S

ν∈I_n^∗S_g,ν ⊆S

ν∈I_n^∗op_g(Og,ν)
∨ S_ν∈I∗

nS_g,ν ⊇S

ν∈I_n^∗¬ op_g(Kg,ν)

⇒ S

ν∈I_n^∗S_g,ν ⊆ op_g S

ν∈I_n^∗O_g,ν

∨ S_ν∈I∗

nS_g,ν ⊇¬ op_g S

ν∈I_n^∗K_g,ν

But,S

ν∈I_n^∗O_g,ν ∈ T_g and S

ν∈I_n^∗K_g,ν ∈ ¬T_g. Hence,S

ν∈I_n^∗S_g,ν ∈ g-ST_g.

Theorem 3.9. If S_g,1, S_g,2, . . ., S_g,n ∈ g-ST_g are n ≥ 1 g-T_g-sets of a class g-ST_g in a T_g-space T_g, then

T

ν∈I_n^∗S_g,ν ∈ g-ST_g
∨ T_ν∈I∗

nS_g,ν ∈ g-ST/ _g

(18) Proof. Because, Sg,1, S_g,2, . . ., S_g,n ∈ g-ST_g by hypothesis, the trueness of T_ν∈I∗

nS_g,ν ∈ g-ST_g and T

ν∈I_n^∗S_g,ν∈ g-ST/ _g evidently depend on the following property:

^

ν∈I_n^∗

S_g,ν ⊆ op_g(O_g,ν)
∨ S_g,ν ⊇ ¬ op_g(K_g,ν)

where (Og,ν, Kg,ν) ∈ Tg× ¬T_gfor every ν ∈ I_n^∗. Furthermore, because the g-T_g-set-theoretic operations concern finite intersections, it suffices to prove the theorem for n = 2. Set the first property preceding

∨ to P (ν) and that following ∨ to Q (ν). Then, its decomposition gives V

ν∈I₂^∗ P (ν) ∨ Q (ν)

= V

ν∈I₂^∗P (ν)
∨ V_ν∈I∗ 2 Q (ν)

= P (1) ∧ Q (2)
∨ P (2) ∧ Q (1)

If S_g,1, S_g,2 ∈ g-ST_g are both g-T_g-open sets thenV

ν∈I₂^∗P (ν) is true, and if they are both g-T_g-closed sets thenV

ν∈I₂^∗Q (ν) is true. In these two cases,T

ν∈I₂^∗S_g,ν ∈ g-ST_g. Because, in general, there does not necessarily exists g-T_g-set which is simultaneously g-T_g-open and g-T_g-closed, both P (1) ∧ Q (2) and P (2) ∧ Q (1) are untrue; thus,T

ν∈I₂^∗S_g,ν ∈ g-ST/ _g.

Theorem 3.10. Let S_g ⊂ T_g be a T_g-set and let op_g(·) ∈ L_gΩ be a g-operator in a T_g-space. If S_g∈ g-ST_g, then

op_g(S_g) ∈ g-ST_g
∨ ¬ op_g(S_g) ∈ g-ST_g

(19) Proof. Let Sg ∈ g-ST_g. Then, S_g⊆ op_g(O_g)
∨ S_g ⊇ ¬ op_g(K_g)

for some pair (O_g, K_g) ∈ T_g × ¬T_g of Tg-open and Tg-closed sets relative to Tg. Consequently, op_g(Sg) ⊆ op_g◦ op_g(Og) or

¬ op_g(S_g) ⊇ ¬ op_g◦¬ op_g(K_g). But, op_g◦ op_g(O_g) ⊆ op_g(O_g) and ¬ op_g◦¬ op_g(K_g) ⊇ ¬ op_g(K_g).

Thus, there follows that op_g(S_g) ⊆ op_g(O_g) or ¬ op_g(S_g) ⊇ ¬ op_g(K_g). Hence, op_g(S_g) ∈ g-ST_g or

¬ op_g(Sg) ∈ g-ST_g.

Proposition 3.11. Let S_g∈ g-ST_g in a T_g-space T_g and suppose the logical statement (∃Rg⊂ T_g)

R_g⊆ op_g(Sg)
∨ R_g⊇ ¬ op_g(Sg)


(20) holds, then R_g∈ g-ST_g.

Proof. Let there exists a Tg-set Rg ⊂ T_g such that Rg ⊆ op_g(Sg) or Rg ⊇ ¬ op_g(Sg). But Sg ∈ g-ST_g implies op_g(S_g) ∈ g-ST_g or ¬ op_g(S_g) ∈ g-ST_g. Thus, R_g∈ g-ST_g.

Corollary 3.12. Let Tg be a Tg-space. If g-ST_g = g-OT_g ∪ g-KT_g denotes a class of g-T_g-open and g-T_g-closed sets, and ST_g = OT_g ∪ KT_g denotes a class of T_g-open and T_g-closed sets, then

g-ST_g ⊇ g-OT_g ∪ g-KT_g ⊇ OT_g ∪ KT_g ⊇ ST_g

(21) An important remark should be pointed out at this stage.

Remark 3.13. The converse of the statement “if S_g∈ ST_g then S_g∈ g-ST_g” is obviously untrue.

Because, the negation of this statement gives

S_g ∈ ST_g
∧ ¬ S_g∈ g-ST_g

which is an untrue statements.

Theorem 3.14. Let T_g be a T_g-space. If S_g⊂ T_g, then

S_g∈ g-ST_g ⇔ S_g ⊆ op_g◦¬ op_g(Sg)
∨ S_g⊇ ¬ op_g◦ op_g(Sg)

(22) Proof.

(⇐) : Let

S_g ⊆ op_g◦¬ op_g(S_g)
∨ S_g⊇ ¬ op_g◦ op_g(S_g)

Then, the substitution of ¬ op_g(Sg) = Og in the logical statement preceding ∨ and op_g(Sg) = Kg in that following ∨ gives Sg ⊆ op_g(Og)
∨ S_g ⊇ ¬ op_g(Kg)
.

(⇒) : Let S_g ∈ g-ST_g. Then, S_g⊆ op_g(O_g)
∨ S_g⊇ ¬ op_g(K_g)
. Consequently, substituting O_g = ¬ op_g(Sg) in the logical statement preceding ∨ and Kg = op_g(Sg) in that following ∨, the required logical statement at once follows, which proves the theorem.

The class g-ST_g forms a g-topology on Ω, which will be denoted by T_g-S.

Theorem 3.15. Let g-ST_g be a given g-class in a T_g-space T_g. Then, the one-valued map T_g-S : g-ST_g −→ g-ST_g forms a g-topology on Ω in the T_g-space.

Proof. By definition, ∅ = opg(∅)
∨ ∅ = ¬ op_g(∅)
. Since, either op_g(∅) ⊆ op_g(Og) or ¬ op_g(∅) ⊇

¬ op_g(K_g) holds, where O_g, K_g⊂ T_g, respectively, are some T_g-open and T_g-closed sets in T_g, it follows that ∅ ∈ g-ST_g and, hence, T_g-S(∅) = ∅. Let S_g∈ g-ST_g. Then, since g-ST_g ⊆ g-ST_g, it follows that Sg is a superset of Tg-S(Sg). Hence, Tg-S(Sg) ⊆ Sg. Let Sg,1, Sg,2, . . . be Tg-sets satisfying, for every ν ∈ I_∞^∗ , S_g,ν. Then, there exist classes O_g,ν ∈ T_g : ν ∈ I_∞^∗ and K_g,ν ∈ ¬T_g : ν ∈ I_∞^∗ , respectively, of T_g-open and T_g-closed sets such that

S

ν∈I∞^∗ S_g,ν ⊆ op_g S

ν∈I∞^∗ O_g,ν

∨ S_ν∈I∗

∞S_g,ν ⊇ ¬ op_g S

ν∈I∞^∗ K_g,ν

a relation established on the following expressions:

S

ν∈I∞^∗ op_g(Og,ν) = op_g S

ν∈I∞^∗ O_g,ν
S

ν∈I∞^∗ ¬ op_g(Kg,ν) = ¬ op_g S

ν∈I∞^∗ K_g,ν
Consequently, S

ν∈I_∞^∗ S_g,ν ∈ g-ST_g, since S_ν∈I∗

∞O_g,ν ∈ T_g is a T_g-open set and S

ν∈I_∞^∗ O_g,ν ∈ ¬T_g is a Tg-closed set. Hence,

T_g-S S

ν∈I_∞^∗ S_g,ν
= S_ν∈I∗

∞T_g-S(S_g,ν)

An immediate consequence of the above theorem is the following corollary.

Corollary 3.16. Let a Tg be a Tg-space. Then, the structure (Ω, Tg-S), where Tg-S : g-ST_g

−→

g-ST_g, is a T_g-space.

To condense the set-builder notation describing the classes g-ST_g and then classify it into sub- classes, predicates must be introduced, and the choice made is to consider the so-called Boolean-valued functions on T_g× T_g∪ ¬T_g× L_gΩ × ⊆, ⊇ , the definition of which are given below.

Definition 3.17. Let (S_g, O_g, K_g) ∈ T_g × T_g× ¬T_g and let op_g(·) ∈ L_gΩ be a g-operator in a T_g-space Tg. The first two predicates

P_g S_g, O_g; op_g(·) ; ⊆
:= ∃O_g, op_g(·)

S_g⊆ op_g(O_g)
P_g S_g, K_g; op_g(·) ; ⊇
:= ∃K_g, ¬ op_g(·)

S_g ⊇ ¬ op_g(K_g)
Pg S_g, Og, Kg; op_g(·) ; ⊆, ⊇
:= P_g S_g, Og; op_g(·) ; ⊆

∨ P_g S_g, K_g; op_g(·) ; ⊇

(23) are called a Boolean-valued functions on T_g× T_g∪ ¬T_g× L_gΩ × ⊆, ⊇ .