Theory of G-T G-Separation Axioms 1. Introduction Whether It Concerns The

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

Theory of g-Tg-Separation Axioms

Khodabocus m. i. and Sookia n. u. h.

Abstract. Several specific types of ordinary and generalized separation axioms of a generalized topological space have been defined and investigated for various purposes from time to time in the literature of topological spaces. Our recent research in the field of a new class of generalized separation axioms of a generalized topological space is reported herein as a starting point for more generalized classes.

Key words and phrases. Generalized topological space, generalized separation axioms, generalized sets, generalized operations

1. Introduction

Whether it concerns the theory of T -spaces or Tg-spaces, the idea of adding a 1 T T Tα or a g-Tα-axiom (with α =( 0, 1, 2,)...) to the axioms for a -space( T = (Ω), ) to obtain a T (α)-space T(α) = Ω, T (α) or a g-T (α)-space g-T(α) = Ω, g-T (α) or, the idea of adding a T or a -T -axiom (with α = 0, 1, 2, ...) to the axioms g,α g g,α ( ) T T T (α) (α) T (α) T (α) for a g-space T( g = (Ω, g)) to obtain a g -space Tg = Ω, g or a g- g - (α) T (α) space g-Tg = Ω, g- g has never played little role in Generalized Topology and Abstract Analysis [17, 19, 25]. Because the defining attributes of a T -space in terms of a collection of T -open or g-T -open sets or a Tg-space in terms of a collection of Tg-open or g-Tg-open sets, respectively, does little to guarantee that the points in the T -space T or the Tg-space Tg are somehow distinct or far apart. The more types and categories of Tα or g-Tα-axioms or g-Tα or g-Tg,α-axioms (with α = 0, 1, 2, ...) are added to the axioms for a T -space or a Tg-space, respectively, the greater the role they will⟨ play in any topological⟩ endeavours [4, 8, 10, 18, 23]. For instance, for a sequence g-Tg,α, g-Tg,β, g-Tg,γ (with α, β, γ = 0, 1, 2, ...), the g-Tg,α, g-Tg,β and g-Tg,γ -axioms can be arranged in increasing order of strength in the sense that g-Tg,γ implies g-Tg,β and the latter implies g-Tg,α. In the literature of T -spaces and Tg-spaces, respectively, several classes of Tα, T T g-Tα-axioms, founded upon the concepts of , g- -open sets, and Tg,α, g-Tg,α- axioms (with α = 0, 1, 2, ...), founded upon the concepts of Tg, g-Tg-open sets, have been introduced and studied [1, 9, 11, 12, 13, 20, 22]. The Tα-axioms called TKolmogorov,TFréchet,THausdorff ,TRegular, and TNormal-axioms (shortly, TK,TF, TH,TR, and TN), founded upon the concepts of T -open, closed sets, are four classical examples, among others, which have gained extensive studies [27]. The

1 Notes to the reader: The notations Tα-axiom and g-Tα-axiom (with α = 0, 1, 2, ...), founded upon the notions of T -open and g-T -open sets, respectively, designate an ordinary and T T a generalized separation axioms for a -space T = (Ω, ); the notations Tg,α-axiom and g-Tg,α- axiom (with α = 0, 1, 2, ...), founded upon Tg-open and g-Tg-open sets, respectively, designate an ordinary and a generalized separation axioms for a Tg-space T = (Ω, Tg). 1

2 KHODABOCUS M. I. AND SOOKIA N. U. H.

g-Tα-axioms called generalized Tα,Sβ-axioms (with α = 0, 1, 2; β = 1, 2), founded upon g-T -open sets instead of T -open sets, are five examples of generalized Tα- axioms which have been discussed in the paper of [5]; the g-Tg,α-axioms called generalized T α -axioms (with α = 2, 3, 4), founded upon g-Tg-open sets instead of 8 Tg-open sets, are three examples of generalized Tg,α-axioms which have introduced and studied by [26]. Several other class of Tα, g-Tα-axioms and Tg,α, g-Tg,α-axioms (with α = 0, 1, 2, ...) have also been introduced and discussed in many papers [14, 16, 20, 25, 24, 28]. ⟨ ⟩ In view of the above references, we remark that the quintuple sequence T , { } α α∈Λ where Λ = K, F, H, R, N , is based on the notions of T -open, closed sets.⟨ From⟩ this remark and the conclusion drawn by [5], it is no error to state that g-T ⟨ ⟩ α α∈Λ are based on the notions of g-T -open, closed sets; T on the notions of ⟨ ⟩ g,α α∈Λ T -open, closed sets, and g-T on the notions of g-T -open, closed sets. g g,α α∈Λ ⟨ ⟩ g Thus, the idea of adding a quintuple sequence -T of -T -axioms (with { } g g,α α∈Λ g g,α T Λ = K, F, H, R, N ), founded upon a new class of g- g-open, closed sets,⟨ to the T T (α) (axioms for)⟩ a g-space Tg = (Ω, g) to obtain a corresponding sequence g-Tg = T (α) T (α) Ω, g- g α∈Λ of g- g -spaces might be an interesting subject of inquiry. T Hitherto, the introduction of several types of Tα and g-Tα-axioms in -spaces T and Tg,α and g-Tg,α-axioms (with α = 0, 1, 2, ...) in g-spaces have contributed extensively to the geometrical specifications of T -spaces and Tg-spaces. However, despite these contributions⟨ ⟩ not a single work has been devoted to the generalization T of the sequence Tα α∈Λ in terms of the notions of g- -open, closed sets. With this view in mind, the idea therefore suggests itself, of introducing the generalized versions of the Kolmogorov, Fréchet, Hausdorff, Regular and Normal separation axioms in terms of the notions of g-T -open, closed sets in a Tg-space, adequate for the obtention of g-T -spaces in this direction. In this paper, we attempt to make a contribution to such a development by introducing a new theory, called Theory of g-Tg-Separation⟨ ⟩ Axioms, in which it is presented the generalized version of the sequence Tα α∈Λ in terms of the notions of g-T -open, closed sets, discussing the fundamental properties and giving charac- terizations of its elements, on this ground and with respect to existing works. The paper is organised as follows: In Sect. 2, preliminary notions are described Sect. T in 2.1 and the main results of the g-Tg,α-axioms in a g-space are reported in Sect. 3. In Sect. 4, the establishment of the various relationships between Sect. these g-Tg,α-axioms are discussed in 4.1. To support the work, a nice T Sect. application of the g-Tg,α-axioms in a g-space is presented in 4.2. Finally, Sect. 4.3 provides concluding remarks and future directions of the g-Tg,α-axioms in a Tg-space.

2. Theory 2.1. Preliminaries. Though foreign terms are neatly defined in complementary chapters (see our papers on theories of g-Tg-sets and g-Tg-maps), we thought it necessary to recall some basic definitions and notations of most essential concepts presented in those chapters. The set U represents the universe of discourse, fixed within the framework of the theory of g-Tg-separation axioms and containing as elements all sets (T , g-T , T, Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

THEORY OF g-Tg-SEPARATION AXIOMS 3 { T T 0 def ∈ N0 g-T-sets;} g, g- g, Tg, g-Tg-sets) considered in this theory, and In = ν : ν ≤ n ; index sets I0 , I∗, I∗ are deﬁned in an analogous way. Granted Ω ⊂ U, { ∞ n ∞ } def ∗ P (Ω) = Og,ν ⊆ Ω: ν ∈ I∞ denotes the family of all subsets Og,1, Og,2, ..., T P → P T ∅ ∅ of Ω. A one-valued map(∪ of the type) ∪g : (Ω) (Ω) satisfying g ( ) = , T (O ) ⊆ O , and T ∗ O = ∗ T (O ) is called a g-topology on Ω, g g g g ν∈I∞ g,ν ν∈I∞ g g,ν def T T and⟨ the⟩ structure Tg = (Ω, g) is called{ a g-space, on} which a quintuple sequence g-Tg,α α∈Λ of g-Tg,α-axioms (with Λ = K, F, H, R, N ) will be discussed [7, 6, 21]. The operator clg : P (Ω) → P (Ω) carrying each Tg-set Sg ⊂ Tg,Ω into its closure clg (Sg) = Tg − intg (Tg \Sg) ⊂ Tg is termed a g-closure operator and the operator intg : P (Ω) → P (Ω) carrying each Tg-set Sg ⊂ Tg into its interior intg (Sg) = Tg − clg (Tg \Sg) ⊂ Tg is called a g-interior operator. Let { : P (Ω) → P (Ω) denotes the absolute complement with respect to the underlying set Ω ⊂ U, and let Sg ⊂ Tg be any Tg-set. The classes { } def Tg = Og ⊂ Tg : Og ∈ Tg , { } def (2.1) ¬Tg = Kg ⊂ Tg : {Λ (Kg) ∈ Tg ,

respectively, denote the classes of all Tg-open and Tg-closed sets relative to the g-topology Tg, and the classes { } Csub [S ] def= O ∈ T : O ⊆ S , Tg g g g g g { } (2.2) Csup [S ] def= K ∈ ¬T : K ⊇ S , ¬Tg, g g g g g

respectively, denote the classes of Tg-open subsets and Tg-closed supersets (comple- ments of the Tg-open subsets) of the Tg-set Sg ⊂ Tg relative to the g-topology Tg. To this end, the g-closure and the g-interior of a Tg-set Sg ⊂ Tg in a Tg-space [2] deﬁne themselves as ∪ ∩ def def (2.3) intg (Sg) = Og, clg (Sg) = Kg. sub sup O ∈C [S ] Kg∈C [Sg] g Tg g ¬Tg

Throughout this paper, the composition operators clg ◦ intg (·), intg ◦ clg (·), and clg ◦ intg ◦ clg (·), respectively, stand for the functionals clg (intg (·)), intg (clg (·)), and clg (intg (clg (·))); other composition operators are defined similarly. Further- more, the backslash Tg \Sg refers to the set-theoretic difference Tg − Sg. The P → P P mapping opg : (Ω) (Ω) is called a g-operation on (Ω) if the following statements hold: { } { } { } ∀Sg ∈ P (Ω) \ ∅ , ∃ (Og, Kg) ∈ Tg \ ∅ × ¬Tg \ ∅ : ( ) ( ) ( ) ( ) ∅ ∅ ∨ ¬ ∅ ∅ S ⊆ O ∨ S ⊇ ¬ K (2.4) opg ( ) = opg ( ) = , g opg ( g) g opg ( g) , ¬ P → P P where opg : (Ω) (Ω) is called the ”complementary{ } g-operation” on (Ω) and, for all T -sets S , S , S ∈ P (Ω) \ ∅ , the following axioms are satisfied: g ( g g,ν g),µ ( ) • Ax. i. S ⊆ O ∨ S ⊇ ¬ K ( g opg ( g) g )opg(( g) , ) • Ax. ii. S ⊆ ◦ O ∨ ¬ S ⊇ ¬ ◦¬ K op( g ( g) opg opg ( g) opg ( g) )opg( opg ( g) , • Ax. iii. S ⊆ S → O ⊆ O ∨ S ⊆ S ← g,ν g,µ ) opg ( g,ν ) opg ( g,µ) g,µ g,ν ¬ K ⊇ ¬ K opg ( g,µ) opg ( g,ν ) , Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

4 KHODABOCUS M. I. AND SOOKIA N. U. H. ( (∪ ) ∪ ) ( (∪ ) • Ax. iv. S ⊆ O ∨ ¬ S ⊇ ∪ opg σ=ν,µ) g,σ σ=ν,µ opg ( g,σ) opg σ=ν,µ g,σ ¬ op (K ) , σ=ν,µ g g,σ { } T O O O ∈ T \ ∅ T K K for some g-open sets g, g,ν[, ] g,µ [ g] [ and] g-closed sets g, g,ν , K ∈ ¬T L Lω × Lκ g,µ g [3, 15]. The class g Ω = g Ω g Ω , where [ ] { ( ) } L def · · ¬ · ∈ 0 × 0 (2.5) g Ω = opg,νµ ( ) = opg,ν ( ) , opg,µ ( ) :(ν, µ) I3 I3

in the Tg-space Tg, stands for the class of all possible g-operators and their complementary g-operators in the Tg-space Tg. Its elements are deﬁned as: [ ] { } · ∈ Lω def · · · · opg ( ) g Ω = opg,0 ( ) , opg,1 ( ) , opg,2 ( ) , opg,3 ( ) { } = intg (·) , clg ◦ intg (·) , intg ◦ clg (·) , clg ◦ intg ◦ clg (·) ; [ ] { } ¬ · ∈ Lκ def ¬ · ¬ · ¬ · ¬ · opg ( ) g Ω = opg,0 ( ) , opg,1 ( ) , opg,2 ( ) , opg,3 ( ) { } (2.6) = clg (·) , intg ◦ clg (·) , clg ◦ intg (·) , intg ◦ clg ◦ intg (·) . S ⊂ T A Tg-set g Tg in a g-space is called a g-Tg-set if and only if there exist a pair[ ] O K ∈ T ×¬T T T · ∈ L ( g, g) g g of g-open and g-closed sets, and a g-operator opg ( ) g Ω such that the following statement holds: [ (( ) ( ))] ∃ ∈ S ∧ S ⊆ O ∨ S ⊇ ¬ K (2.7) ( ξ) (ξ g) g opg ( g) g opg ( g) .

The g-Tg-set Sg ⊂ Tg is said to be of category ν if and only if it belongs to the following class of g-ν-Tg-sets: [ ] { ( ) def S ⊂ ∃O K · g-ν-S Tg = g Tg : g, g, opg,ν ( ) [( ) ( )]} (2.8) S ⊆ op (O ) ∨ S ⊇ ¬ op (K ) . g g,ν g g g,ν g [ ] It is called a g-ν-Tg-open set if it satisfies the first property[ in]g-ν-S Tg and a g-ν-Tg-closed set if it satisfies the second property in g-ν-S Tg . The classes of g-ν-Tg-open and g-ν-Tg-closed sets, respectively, are defined by [ ] { ( )[ ]} def S ⊂ ∃O · S ⊆ O g-ν-O Tg = g Tg : g, opg,ν ( ) g opg,ν ( g) , [ ] { ( )[ ]} def S ⊂ ∃K · S ⊇ ¬ K g-ν-K Tg = g Tg : g, opg,ν ( ) g opg,ν ( g) . (2.9) From these classes, the following relation holds: [ ] ∪ [ ] g-S Tg = ν∈I0 g-ν-S Tg ∪ 3 ( [ ] [ ]) ∪ = ν∈I0 g-ν-O Tg g-ν-K Tg (∪ 3 [ ]) (∪ [ ]) ∪ = ν∈I0 g-ν-O Tg ν∈I0 g-ν-K Tg [ 3 ] [ ] 3 (2.10) = g-O Tg ∪ g-K Tg . When the subscript g are omitted in almost all symbols of the above definitions, very similar definitions are derived but in a T -space. 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 (·) ∈ L Ω such that the following statement holds: [ ( )] (2.11) (∃ξ) (ξ ∈ S) ∧ (S ⊆ op (O)) ∨ (S ⊇ ¬ op (K)) . Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

THEORY OF g-Tg-SEPARATION AXIOMS 5

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: [ ] { def S ⊂ ∃O K · g-ν-S T = T :( , , opν ( )) [ ]} S ⊆ O ∨ S ⊇ ¬ K (2.12) ( opν ( )) ( opν ( )) . [ ] It is called a g-ν-T-open set if it satisfies the first property[ ] in g-ν-S T and a g-ν-T- closed set if it satisfies the second property in g-ν-S T . The classes of g-ν-T-open and g-ν-T-closed sets, respectively, are defined by [ ] { [ ]} def S ⊂ ∃O · S ⊆ O g-ν-O T = T :( , opν ( )) opν ( ) , [ ] { [ ]} def S ⊂ ∃K · S ⊇ ¬ K (2.13) g-ν-K T = T :( , opν ( )) opν ( ) . The following relations are immediate consequences of the above definitions: [ ] ∪ [ ] g-S T = ν∈I0 g-ν-S T ∪ 3 ( [ ] [ ]) ∪ = ν∈I0 g-ν-O T g-ν-K T (∪ 3 [ ]) (∪ [ ]) ∪ = ν∈I0 g-ν-O T ν∈I0 g-ν-K T [ 3] [ ] 3 (2.14) = g-O T ∪ g-K T .

The classes O [Tg] and K [Tg] denote the families of Tg-open and Tg-closed sets, respectively, in Tg, with S [Tg] = O [Tg] ∪ K[Tg]; 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 regard to the above descriptions, by a g-Tg-open set and a g-Tg-closed set are T O ∈ T T K ∈ ¬T O ⊆ O meant a g-open set g g and a g-closed set g g satisfying g opg ( g) K ⊇ ¬ K T and g opg ( g), respectively. Likewise, by a g- g-open set of category ν and a g-Tg-closed set of category ν are meant a Tg-open set Og ∈ Tg and a Tg-closed K ∈ ¬T O ⊆ O K ⊇ ¬ K T set g g satisfying g opg,ν ( g) and g opg,ν ( g), respectively; g- g- sets of category ν will be called g-ν-Tg-sets. We are now in a position to present a carefully chosen set of terms used in the theory of g-Tg-separation axioms in T g-spaces. ⟨ ⟩ Agreed to let g-T denote a sequence of g-T -axioms, indexed by the { g,α} α∈Λ g,α ⟨ def (α) (set Λ = )⟩K, F, H, R, N , throughout the present chapter, the sequence g-Tg = Ω, g-T (α) will stand for the resulting sequence of g-T (α)-spaces, obtained g α∈Λ ( ) ⟨ ⟩ g T T after endowing a g-space Tg = Ω, g with g-Tg,α α∈Λ. Hence, the deﬁnition follows. Definition 2.1 (g-T (α)-Space). A T -space T = (Ω, T ) endowed with a g-T - g g ( g ) g g,α T (α) (α) def T (α) axiom is called a g- g -space g-Tg = Ω, g- g . ⟨ ⟩ T The elements of g-Tg,α α∈Λ concern the separation of points, points from g- g- open sets, and g-Tg-open sets from each other. They are nicely discussed through T T the notions of pairwise disjoint points and g- g-sets in a g-space Tg. We let O K ∈ T × ¬T T T ( g,ξ, g,ξ) g g( denote a pair) of g-open and g-closed sets containing the ∈ O K ∈ T × ¬T T point ξ Tg and let g,Sg , g,Sg g g denote either a pair of g-open and Tg-closed subsets or supersets of the set Sg ∈ Tg, and consider the following deﬁnition. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

6 KHODABOCUS M. I. AND SOOKIA N. U. H.

Definition 2.2 (ξ, Sg-Pairwise Disjoint). Let Tg = (Ω, Tg) be a Tg-space. For some σ ≥ 0 and Sg ⊇ ∅, the families { } def ∈ × N ≥ g-FP [σ] = (ξ, ζ) Tg Tg : g (ξ, ζ) σ , { ∩ ( ) } S def O O ∈ T × T O ⊆ S g-ν-FO [ g] = ( g,ξ, g,ζ ) g g : λ=ξ,ζ opg,ν g,λ g , { ∩ ( )} S def K K ∈ ¬T × ¬T S ⊇ ¬ K g-ν-FK [ g] = ( g,ξ, g,ζ ) g g : g λ=ξ,ζ opg,ν g,λ , (2.15) T T respectively, denote the collections of pairwise points, and g- g-open and g- g- closed sets of category ν in Tg. They form the collections of pairwise distinct points, T T and pairwise disjoint g- g-open and g- g-closed sets of category ν whenever σ > 0 and Sg = ∅, respectively. ⟨ ⟩ S S Granted g-FP [σ], g-ν-FO [ g], and g-ν-FK [ g], the elements of g-ν-Tg,α α∈Λ may well be stated as thus: ⟨ ⟩ Definition . T T 2.3 ( g-ν-Tg,α α∈Λ-Axioms) Let Tg = (Ω, g) be a g-space and let S ⊆ T × T S ⊆ ¬T × ¬T g-FP [σ], and g-ν-FO [ g] g g and g-ν-FK [ g] g g be given, where σ ≥ 0 and Sg ⊇ ∅. Then: • i. -Axiom ∈ g-ν-Tg,K :[ For every] (ξ, ζ) g-FP [σ > 0], there exists a pair O O ∈ S ⊃ ∅ ( g,ξ, g,ζ ) g-ν-FO g such that: [( ( )) ( ( ))] [( ( )) ∈ O ∧ ∈ O ∨ ∈ O ξ opg,ν g,ξ ζ / opg,ν g,ξ ξ / opg,ν g,ζ ( ( ))] ∧ ∈ O (2.16) ζ opg,ν g,ζ . • ii. -Axiom ∈ g-ν-Tg,F :[ For every] (ξ, ζ) g-FP [σ > 0], there exists a pair O O ∈ S ⊃ ∅ ( g,ξ, g,ζ ) g-ν-FO g such that: [ ] [ ] ∈ O ∧ ∈ O (2.17) (ξ, ζ) ×λ=ξ,ζ opg,ν ( g,λ) (ξ, ζ) / ×λ=ζ,ξ opg,ν ( g,λ) . • iii. -Axiom ∈ g-ν-Tg,H :[ For] every (ξ, ζ) g-FP [σ > 0], there exists a pair O O ∈ ∅ ( g,ξ, g,ζ ) g-ν-FO such that: [ ] [ ] ∈ O ∧ ∈ O (2.18) ξ opg,ν ( g,ξ) ζ opg,ν ( g,ζ ) . • iv. -Axiom ∈ K K ∈ g-ν[-T] g,R : For every (ξ, ζ) g-FP [σ > 0] and ( g,ξ, g,ζ ) ∅ ∈ K K O O ∈ g-ν-FK[ ] such that (ζ, ξ) / ( g,ξ, g,ζ ), there exists a pair ( g,ξ, g,ζ ) ∅ g-ν-FO such that: [( ) ( )] [( ¬ K ⊂ O ∧ ∈ O ∨ ¬ K opg,ν ( g,ξ) opg,ν ( g,ξ) ζ opg,ν ( g,ζ ) opg,ν ( g,ζ ) ) ( )] ⊂ O ∧ ∈ O (2.19) opg,ν ( g,ζ ) ξ opg,ν ( g,ξ) . • v. -Axiom K K ∈ ∅ g-ν-Tg,N : For every ( g,ξ, g,ζ ) g-ν-FK [ ], there exists a pair O O ∈ ∅ ( g,ξ, g,ζ ) g-ν-FO [ ] such that: [ ] [ ] O ⊃ ¬ K ∧ O ⊃ ¬ K (2.20) opg,ν ( g,ξ) opg,ν ( g,ξ) opg,ν ( g,ζ ) opg,ν ( g,ζ ) . ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ def ∨ Granted g-ν-T , we form g-T = ∈ 0 g-ν-T , and de- g,α α∈Λ g,α α∈Λ ν I3 g,α α∈Λ ﬁne the g-Tg,K, g-Tg,F, g-Tg,H, g-Tg,R, and g-Tg,N-axioms as thus. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

THEORY OF g-Tg-SEPARATION AXIOMS 7 ⟨ ⟩ Definition 2.4 ( -T -Axioms). Let = (Ω, T ) be a T -space and let g ∪g,α α∈Λ Tg g ∪ g g-F [σ], g-F [Sg] = ∈ 0 g-ν-F [Sg] ⊆ Tg × Tg, g-F [Sg] = ∈ 0 g-ν-F [Sg] ⊆ P O ν I3 O K ν I3 K ¬Tg × ¬Tg be given, where σ ≥ 0 and Sg ⊇ ∅. Then: • i. -Axiom ∈ O O ∈ g-T[g,K ] : For every (ξ, ζ) g-FP [σ > 0], there exists ( g,ξ, g,ζ ) g-F S ⊃ ∅ such that: O g[( ( )) ( ( ))] [( ( )) ∈ O ∧ ∈ O ∨ ∈ O ξ opg g,ξ ζ / opg g,ξ ξ / opg g,ζ ( ( ))] ∧ ∈ O (2.21) ζ opg g,ζ . • ii. -Axiom ∈ g-Tg,F :[ For every] (ξ, ζ) g-FP [σ > 0], there exists a pair (O , O ) ∈ g-F S ⊃ ∅ such that: [g,ξ g,ζ O g ] [ ] ∈ O ∧ ∈ O (2.22) (ξ, ζ) ×λ=ξ,ζ opg ( g,λ) (ξ, ζ) / ×λ=ζ,ξ opg ( g,λ) . • iii. -Axiom ∈ g-Tg,H [:] For every (ξ, ζ) g-FP [σ > 0], there exists a pair (O , O ) ∈ g-F ∅ such that: g,ξ g,ζ [ O ] [ ] ∈ O ∧ ∈ O (2.23) ξ opg ( g,ξ) ζ opg ( g,ζ ) . • iv. -Axiom ∈ K K ∈ [g-T] g,R : For every (ξ, ζ) g-FP [σ > 0] and ( g,ξ, g,ζ )[ ] ∅ ∈ K K O O ∈ ∅ g-FK such that (ζ, ξ) / ( g,ξ, g,ζ ), there exists ( g,ξ, g,ζ ) g-FO such that: [( ) ( )] [( ¬ K ⊂ O ∧ ∈ O ∨ ¬ K opg ( g,ξ) opg ( g,ξ) ζ opg ( g,ζ ) opg ( g,ζ ) ) ( )] ⊂ O ∧ ∈ O (2.24) opg ( g,ζ ) ξ opg ( g,ξ) . • v. -Axiom K K ∈ ∅ g-Tg,N : For every ( g,ξ, g,ζ ) g-FK [ ], there exists a pair (O , O ) ∈ g-F [∅] such that: [g,ξ g,ζ O ] [ ] O ⊃ ¬ K ∧ O ⊃ ¬ K (2.25) opg ( g,ξ) opg ( g,ξ) opg ( g,ζ ) opg ( g,ζ ) .

In the following sections, the main results of the theory of g-Tg-maps are presented.

3. Main Results ( ) T T T (K) A necessary and( suﬃcient) condition for a g-space Tg = Ω, g to be a g- g - (K) T (K) space g-Tg = Ω, g- g may be given in terms of the complementary g-operator ¬ P → P { } { } ⊂ × opg : (Ω) (Ω) and any pairs ( ξ , ζ ) Tg Tg of unit sets. ( ) Theorem . T T T (K) (K) ( )3.1 A g-space Tg = Ω, g is said to be a g- g -space g-Tg = T (K) Ω, g- g if and only if the following condition holds: ¬ { } ̸ ¬ { } ∀ ∈ (3.1) opg ( ξ ) = opg ( ζ ) (ξ, ζ) g-FP [σ > 0] . Proof. T T (K) (K) Necessity. Let the g-space Tg be a g- g -space[ g-]Tg . Then, for every (ξ, ζ) ∈ g-F [σ > 0], there exists (O , O ) ∈ g-F S ⊃ ∅ such that: P [( ( )) ( g,ξ g,ζ( ))]O [(g ( )) ∈ O ∧ ∈ O ∨ ∈ O ξ opg g,ξ ζ / opg g,ξ ξ / opg g,ζ ( ( ))] ∧ ∈ O ζ opg g,ζ . Consequently, [( ( ( ))) ( ( ( )))] [( ( ( ))) ∈ { O ∧ ∈ { O ∨ ∈ { O ξ / opg g,ξ ζ opg g,ξ ξ opg g,ζ ( ( ( )))] ∧ ∈ { O ζ / opg g,ζ , Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

8 KHODABOCUS M. I. AND SOOKIA N. U. H. ( ( )) ( ( )) [ ] { O { O ∈ implying opg g,ξ , opg g,ζ g-K Tg , respectively, are g-Tg-closed sets ∈ ∈ K K ∈ ¬T × ¬T containing( (ζ T))g and ξ ( Tg.) Thus, there( ( exists)) ( g,ξ, g(,ζ ) ) g g such { O ⊇ ¬ K { O ⊇ ¬ K that opg g,ξ (opg g,ζ )and ( op(g g,ζ)) opg ( g,ξ ). But, for every ∈ { } { { } ⊇ { { } ⊇ { O ¬ K ⊇ ¬ { } λ ξ, ζ , ( λ ) opg ( λ ) opg g,λ and opg g,λ opg ( λ ). { { } ⊇ ¬ { } { { } ⊇ ¬ { } { { } ̸ { { } Therefore, ( ξ ) opg ( ζ ) and ( ζ ) opg ( ξ ). Since, ( ξ ) = ( ζ ), ¬ { } ̸ ¬ { } it follows that opg ( ξ ) = opg ( ζ ). ¬ { } ̸ ¬ { } Suﬃciency. Conversely, suppose the condition opg ( ξ ) = opg ( ζ ) holds for every (ξ, ζ) ∈ g-F [σ > 0]. Then there exists η ∈ Tg such that [( P ) ( ( ))] [( ( )) ∈ ¬ { } ∧ ∈ ¬ { } ∨ ∈ ¬ { } η opg ( ξ ) η / opg ζ η / opg ξ ( )] ∧ ∈ ¬ { } η opg ( ζ ) . [ ] [ ( )] ∈ ¬ { } ∧ ∈ ¬ { } If ξ opg ( ζ ) ζ opg ξ , then [ ] [ ] ¬ { } ⊆ ¬ { } ∧ ¬ { } ⊆ ¬ { } opg ( ξ ) opg ( ζ ) opg ( ζ ) opg ( ξ ) . Consequently, [( ) ( ( ))] [( ( )) ∈ ¬ { } ∧ ∈ ¬ { } ∨ ∈ ¬ { } η opg ( ζ ) η / opg ζ η / opg ξ ( )] ∧ ∈ ¬ { } η opg ( ξ ) . [ ] [ ( )] ∈ ¬ { } ∧ ∈ ¬ { } This is a contradiction; hence, ξ / opg ( ζ ) ζ / opg ξ , implying [ ( )] [ ( ( ))] ∈ { ¬ { } ∧ ∈ { ¬ { } ξ opg ( ζ ) ζ opg ξ . ( ) ( ( )) [ ] { ¬ { } { ¬ { } ∈ Since opg ( ζ ) , opg ξ g-O Tg , respectively, are g-Tg-open sets ∈ ∈ O O ∈ T × T containing( ξ ) Tg and( ζ ) Tg, there( exists) ( g,ξ, g(,ζ ) ) g g such that { ¬ { } ⊆ O { ¬ { } ⊆ O opg ( ζ ) opg g,ξ and opg ( ξ ) [opg g,ζ] . Hence, for every ∈ O O ∈ S ⊃ ∅ (ξ, ζ) g-FP [σ > 0], there exists ( g,ξ, g,ζ ) g-FO g such that: [( ( )) ( ( ))] [( ( )) ∈ O ∧ ∈ O ∨ ∈ O ξ opg g,ξ ζ / opg g,ξ ξ / opg g,ζ ( ( ))] ∧ ∈ O ζ opg g,ζ . ( ) ( ) T T (K) (K) T (K) Therefore, Tg = Ω, g is a g- g -space g-Tg = Ω, g- g ; this completes the proof of the theorem. Q.e.d. ( ) T T T (F) A necessary( and suﬃcient) condition for a g-space Tg = Ω, g to be a g- g - (F) T (F) space g-Tg = Ω, g- g may be given in terms of the complementary g-operator ¬ P → P { } ⊂ opg : (Ω) (Ω) and a unit set ξ Tg. ( ) Theorem . T T T (F) (F) ( )3.2 A g-space Tg = Ω, g is said to be a g- g -space g-Tg = T (F) Ω, g- g if and only if the following condition holds: { } ⊇ ¬ { } ∀ ∈ (3.2) ξ opg ( ξ ) ξ Tg. Proof. T T (F) (F) Necessity. Let the g-space Tg be a g- g -space[ g-]Tg . Then, for every ∈ O O ∈ S ⊃ ∅ (ξ, ζ) g-FP [σ > 0], there exists ( g,ξ, g,ζ ) g-FO g such that: [ ] [ ] ∈ O ∧ ∈ O (ξ, ζ) ×λ=ξ,ζ opg ( g,λ) (ξ, ζ) / ×λ=ζ,ξ opg ( g,λ) . Consequently, [ ( )] [ ( )] ∈ { O ∧ ∈ { O (ξ, ζ) / ×λ=ξ,ζ opg ( g,λ) (ξ, ζ) ×λ=ζ,ξ opg ( g,λ) . Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

THEORY OF g-Tg-SEPARATION AXIOMS 9 ( ) [ ] ∈ { } { O ∈ Since, for every λ ξ, ζ , opg ( (g,λ) g-K) Tg is a g-Tg-closed set,( there exists) K K ∈ ¬T ×¬T { O ⊇ ¬ K { O ⊇ ( g,ξ, g,ζ ) g g such that opg ( g,ξ() ) opg ( g,ζ ) and opg ( g,ζ ) ¬ op (Kg,ξ). But, for every λ ∈ {ξ, ζ}, ¬ op Kg,λ ⊇ ¬ op ({λ}). Therefore g [ ] [ g g ] ∈ ¬ { } ∧ ∈ ¬ { } (ξ, ζ) ×λ=ξ,ζ opg ( λ ) (ξ, ζ) / ×λ=ζ,ξ opg ( λ ) . ∈ { } ⊇ ¬ { } Hence, for every ξ Tg, ξ opg ( ξ ). { } ⊇ ¬ { } Suﬃciency. Conversely, suppose the condition ξ opg ( ξ ) holds for every ξ ∈ T . Let (ξ, ζ) ∈ T × T such that ξ ≠ ζ. Then g [ g g ( )] [ ( )] (ξ, ζ) ∈/ × { {λ} ∧ (ξ, ζ) ∈× { {λ} . λ=(ξ,ζ ) ( ) λ=ζ,ξ ( ) [ ] ∈ { } { { } ⊆ { ¬ { } { ¬ { } ∈ But, for every λ ξ, ζ , λ opg ( λ ) , and opg ( λ () g-O Tg) is O O ∈ T ×T { ¬ { } ⊆ a g-Tg-open set.( Thus, there exists) ( g,ξ, g,ζ ) g g such that opg ( ξ ) O { ¬ { } ⊆ O opg ( g,ζ ) and opg ( ζ ) opg ( g,ξ). By substitution,[ it thus] follows that, for every (ξ, ζ) ∈ g-F [σ > 0], there exists (Og,ξ, Og,ζ ) ∈ g-F Sg ⊃ ∅ such that: [ P ] [ O ] ∈ O ∧ ∈ O (ξ, ζ) ×λ=ξ,ζ opg ( g,λ) (ξ, ζ) / ×λ=ζ,ξ opg ( g,λ) . ( ) ( ) T T (F) (F) T (F) Therefore, Tg = Ω, g is a g- g -space g-Tg = Ω, g- g ; this completes the proof of the theorem. Q.e.d. ( ) Proposition . T T (F) (F) T (F) 3.3 If Tg =( (Ω, g) is) a g- g -space g-Tg = Ω, g- g , then it T (K) (K) T (K) is a g- g -space g-Tg = Ω, g- g . ( ) Proof. T T (F) (F) T (F) Let Tg = (Ω, g) be a g- g -space g-Tg = Ω, g-[ g .] Then, for every (ξ, ζ) ∈ g-F [σ > 0], there exists a pair (Og,ξ, Og,ζ ) ∈ g-F Sg ⊃ ∅ such that: [P ] [ O ] (ξ, ζ) ∈× op (Og,λ) ∧ (ξ, ζ) ∈/ × op (Og,λ) . ( λ=(ξ,ζ ))g ( ( )) λ=ζ,ξ g ∈ O ∧ ∈ O Set P (ξ, ζ) = ξ opg g,ξ ζ / opg g,ξ . Then, the above logical statement is equivalent to P (ξ, ζ) ∧ P(ζ, ξ). But, logically, ( ) P(ξ, ζ) ∨ P(ζ, ξ) = P (ξ, ζ) ∨ P(ζ, ξ) ∨ P(ξ, ζ) ∧ P(ζ, ξ) . ∨ ← ∧ Consequently, P (ξ, ζ) P(ζ, ξ) P(ξ, ζ) P(( ζ, ξ), from) which it then follows T T (F) (F) T (F) ∈ that, if Tg = (Ω, g) is a g- g -space g-Tg [= Ω, g]- g , then for every (ξ, ζ) g-F [σ > 0], there exists (Og,ξ, Og,ζ ) ∈ g-F Sg ⊃ ∅ such that: P [( ( )) ( O( ))] [( ( )) ∈ O ∧ ∈ O ∨ ∈ O ξ opg g,ξ ζ / opg g,ξ ξ / opg g,ζ ( ( ))] ∧ ∈ O ζ opg g,ζ , T T (K) (K) (the logical) statement characterising Tg = (Ω, g) as a g- g -space g-Tg = T (K) Q.e.d. Ω, g- g . ( ) T T T (H) A necessary and( suﬃcient) condition for a g-space Tg = Ω, g to be a g- g - (H) T (H) P → space g-Tg = Ω, g- g may be given in terms of the g-operator opg : (Ω) P (Ω), a unit set {ξ} ⊂ Tg, and Tg-closed sets. ( ) Theorem . T T T (H) (H) ( )3.4 A g-space Tg = Ω, g is said to be a g- g -space g-Tg = Ω, g-T (H) if and only if the following conditions hold: g ∩ { } ¬ K ∀ ∈ (3.3) ξ = opg ( g,ζ ) ξ Tg.

Kg,ζ ∈¬Tg Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

10 KHODABOCUS M. I. AND SOOKIA N. U. H.

Proof. T T (H) (H) Necessity. Let the g-space Tg be a g- g -space[ ] g-Tg . Then, for every ∈ O O ∈ ∅ (ξ, ζ) g-FP [σ > 0], there exists ( g,ξ, g,ζ ) g-FO such that: [ ] [ ] ∈ O ∧ ∈ O ξ opg ( g,ξ) ζ opg ( g,ζ ) . Consequently, [ ( )] [ ( )] ∈ { O ∧ ∈ { O ξ / opg ( g,ξ) ζ / opg ( g,ζ ) . ( ) [ ] ∈ { } { O ∈ But, for every λ ξ, ζ , opg ( g,λ) g-K Tg is( a g-Tg-closed) set. Conse- K K ∈ ¬T ×¬T { O ⊇ ¬ K quently,( there exists) ( g,ξ, g,ζ ) g g such that opg ( (g,ξ) ) opg (( g,ζ) { O ⊇ ¬ K ∈ { { } ⊇ ¬ { } and opg (( g,ζ)) opg(( g),ξ). Therefore, the relations ξ ζ op( g )ξ ∈ { { } ⊇ ¬ { } ∈ × { { } ∪and ζ ξ opg ζ are true for all (ξ, ζ) Tg Tg. But, ξ = op (O ) and hence, Og,ζ ∈Tg g g,ζ ( ( )) ∩ ( ) ∩ { } { { { } { O ¬ K ∀ ∈ ξ = ξ = opg ( g,ζ ) = opg ( g,ζ ) ξ Tg.

Og,ζ ∈Tg Kg,ζ ∈¬Tg ∩ Suﬃciency. Conversely, suppose {ξ} = ¬ op (K ) holds for all ξ ∈ T . Kg,ζ ∈¬Tg g g,ζ g K ∈ ¬T ∈ ¬ K ¬ K ∈ Then,[ there] exists a g,ξ g such that ζ / opg ( g,ξ). Since opg[( ]g,ξ) O ∈ g-K Tg is a g-Tg-closed set, there exists a g-Tg-open( set opg )( g,ξ) [ g-O] Tg such ∈ O ⊆ ¬ K { ¬ K ∈ that ξ opg ( g,ξ) opg ( g,ξ). But, since opg ( g(,ξ) g-O T)g is a g[-Tg]- ∈ O { ¬ K ∈ open set containing ζ Tg, it follows that opg ( g,ξ), opg ( g,ξ) g-O Tg ∈ are disjoint g-Tg-open[ ] sets. Thus, for every (ξ, ζ) g-FP [σ > 0], there exists O O ∈ ∅ ( g,ξ, g,ζ ) g-FO such that: [ ] [ ] ∈ O ∧ ∈ O ξ opg ( g,ξ) ζ opg ( g,ζ ) . ( ) ( ) T T (H) (H) T (H) Therefore, Tg = Ω, g is a g- g -space g-Tg = Ω, g- g ; this completes the proof of the theorem. Q.e.d. ( ) Proposition . T T (H) (H) T (H) 3.5 If Tg =( (Ω, g) is) a g- g -space g-Tg = Ω, g- g , then it T (F) (F) T (F) is a g- g -space g-Tg = Ω, g- g . ( ) Proof. T T (H) (H) T (H) Let Tg = (Ω, g) be a g- g -space g-Tg = Ω, g- g [ ]. Then, for every[ ∈ O O ∈ ∅ ∈ (ξ, ζ) g]-FP[[σ > 0], there exists] a pair ( g,ξ, g,ζ ) g-FO[ ] such that[ ξ ] O ∧ ∈ O O O ∈ ∅ ⊂ S ⊃ ∅ opg ( g,ξ) ζ opg ( g,ζ ) . But since ( g,ξ, g,ζ ) g-FO g-FO g and [ ] [ ] ∈ O ∧ ∈ O ξ opg ( g,ξ) ζ opg ( g,ζ ) [( ( )) ( ( ))] [( ( )) ⇔ ∈ O ∧ ∈ O ∧ ∈ O ξ opg g,ξ ζ opg g,ζ ξ / opg g,ζ ( ( ))] ∧ ∈ O ζ / opg g,ξ , ( ) T T (H) (H) T (H) it follows that, if Tg = (Ω, g) is a g- g -space g-Tg = Ω, g- [g , then] for ∈ O O ∈ S ⊃ ∅ every (ξ, ζ) g-FP [σ > 0], there exists a pair ( g,ξ, g,ζ ) g-FO g such that: [ ] [ ] ∈ O ∧ ∈ O (ξ, ζ) ×λ=ξ,ζ opg ( g,λ) (ξ, ζ) / ×λ=ζ,ξ opg ( g,λ) . ( ) T T (H) (H) T (H) T (F) Hence, if Tg =( (Ω, g) is a) g- g -space g-Tg = Ω, g- g , then it is a g- g - (F) T (F) Q.e.d. space g-Tg = Ω, g- g . Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

THEORY OF g-Tg-SEPARATION AXIOMS 11 ( ) T T T (R) A necessary and( suﬃcient) condition for a g-space Tg = Ω, g to be a g- g - (R) T (R) P → space g-Tg = Ω, g- g may be given in terms of the g-operator opg : (Ω) P (Ω), a Tg-closed set Sg ⊂ Tg, and Tg-closed neighbourhood sets. ( ) Theorem . T T T (R) (R) ( )3.6 A g-space Tg = Ω, g is said to be a g- g -space g-Tg = T (R) Ω, g- g if and only if the following condition holds: ∩ ( ) [ ] S ¬ K ∀S ∈ (3.4) g = opg g,Sg g g-K Tg . K ∈¬T g,Sg g Proof. T T (R) (R) Necessity. Let the g-space Tg be a g-[ ]g -space g-Tg . Then, for every ∈ K K ∈ ∅ ∈ K K (ξ, ζ) g-FP [σ > 0] and ( g,ξ[, ]g,ζ ) g-FK such that (ζ, ξ) / ( g,ξ, g,ζ ), there exists (Og,ξ, Og,ζ ) ∈ g-F ∅ such that: [( O ) ( )] [( ¬ K ⊂ O ∧ ∈ O ∨ ¬ K opg ( g,ξ) opg ( g,ξ) ζ opg ( g,ζ ) opg ( g,ζ ) ) ( )] ⊂ O ∧ ∈ O opg ( g,ζ ) ξ opg ( g,ξ) . Consequently, [( ) ( ( ))] [( ¬ K ⊂ O ∧ ∈ { O ∨ ¬ K opg ( g,ξ) opg ( g,ξ) ζ / opg ( g,ζ ) opg ( g,ζ ) ) ( ( ))] ⊂ op (Og,ζ ) ∧ ξ∈ / { op (Og,ξ) . ( ) g [ ] g ∈ { } { O ∈ But, for every λ ξ,( ζ , opg ( g),λ) g-K Tg is a g-Tg-closed( set. Con-) K R K S ∈ ¬T × ¬T { O ⊇ sequently,( there) exists( g, g ,) g, g ( g ) g such that opg ( g,ξ) ¬ op Kg,R and { op (Og,ζ ) ⊇ ¬ op Kg,S . Therefore g [(g g ) g( g ( ))] [( ¬ K ⊂ O ∧ ∈ ¬ K ∨ ¬ K opg ( g,ξ) opg ( g,ξ) ζ / opg g,Sg opg ( g,ζ ) ) ( ( ))] ⊂ O ∧ ∈ ¬ K opg ( g,ζ ) ξ / opg g,Rg . By virtue of this logical statement, it consequently follows that [( ) ( ( ))] ¬ K ⊂ O ∧ ¬ K ⊇ ¬ K opg ( g,ξ) opg ( g,ξ) opg ( g,ξ) opg g,Sg [( ) ( ( ))] ∨ ¬ K ⊂ O ∧ ¬ K ⊇ ¬ K opg ( g,ζ ) opg ( g,ζ ) opg ( g,ξ) opg g,Rg , and, consequently, [ ( ) ] ¬ K ⊆ ¬ K ⊂ O opg g,Sg opg ( g,ξ) opg ( g,ξ) [ ( ) ] ∨ ¬ op Kg,R ⊆ ¬ op (Kg,ζ ) ⊂ op (Og,ζ ) , g( g) g g But, for any Sg ⊂ Tg, ¬ op Kg,S ⊇ Sg. Consequently, [ g ( g ) ] S ⊆ ¬ K ⊆ ¬ K ⊂ O g opg g,Sg opg ( g,ξ) opg ( g,ξ) [ ( ) ] ∨ R ⊆ ¬ K ⊆ ¬ K ⊂ O g opg g,Rg opg ( g,ζ ) opg ( g,ζ ) , ∩ ( ) S ¬ K S ⊂ Hence, g = K ∈¬T opg g,Sg for all g Tg. g,Sg g ∩ ( ) S ¬ K Suﬃciency. Conversely, suppose g = K ∈¬T opg g,Sg holds for all ( g,S)g g S ⊂ ∈ S S ⊆ ¬ K T g Tg, let ξ / g. Then, g opg g,Sg for every g-closed neighbour- K ∈ ¬T S ⊆ K T hood set g,Sg g satisfying g g,Sg . Therefore, there exists a g-closed K ∈ ¬T ∈ K K ∈ ¬T neighbourhood set g,Sg g such that ξ / g,Sg . But, since g,Sg g is T T O S ∈ T a g-closed( neighbourhood) ( set,) there exists( ) a g(-open( set ))g, g ( g such( that)) S ⊂ O ⊂ ¬ K { S ⊃ { O ⊃ { ¬ K g opg g,Sg opg g,Sg , and g opg g,Sg opg g,Sg . Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

12 KHODABOCUS M. I. AND SOOKIA N. U. H. ( ( )) ( ( )) ( ) ∈ { ¬ K S ∈ { O S O S ∩ Because( (ξ )) opg g, g (and(ξ ))opg g, g ( , it follows) that opg g, g { ¬ K S ∅ { O S ∩ ¬ K S ∅ opg g, g = and opg g, g opg g, g = , respectively.[ ] In ∈ K K ∈ ∅ other words, for every (ξ, ζ) g-FP [σ > 0] and ( g[,ξ], g,ζ ) g-FK such that ∈ K K O O ∈ ∅ (ζ, ξ) / ( g,ξ, g,ζ ), there exists ( g,ξ, g,ζ ) g-FO such that: [( ) ( )] [( ¬ K ⊂ O ∧ ∈ O ∨ ¬ K opg ( g,ξ) opg ( g,ξ) ζ opg ( g,ζ ) opg ( g,ζ ) ) ( )] ⊂ O ∧ ∈ O opg ( g,ζ ) ξ opg ( g,ξ) . ( ) ( ) T T (R) (R) T (R) Therefore, Tg = Ω, g is a g- g -space g-Tg = Ω, g- g ; this completes the proof of the theorem. Q.e.d. ( ) Proposition . T T (R) (R) T (R) 3.7 If Tg =( (Ω, g) is) a g- g -space g-Tg = Ω, g- g , then it T (H) (H) T (H) is a g- g -space g-Tg = Ω, g- g . ( ) Proof. T T (R) (R) T (R) Let Tg = (Ω, g) be a g- g -space g-T[ g] = Ω, g- g . Then, for every ∈ K K ∈ ∅ ∈ K K (ξ, ζ) g-FP [σ > 0] and ( g,ξ[, ]g,ζ ) g-FK such that (ζ, ξ) / ( g,ξ, g,ζ ), O O ∈ ∅ there exists ( g,ξ, g,ζ ) g-FO such that: [( ) ( )] [( ¬ K ⊂ O ∧ ∈ O ∨ ¬ K opg ( g,ξ) opg ( g,ξ) ζ opg ( g,ζ ) opg ( g,ζ ) ) ( )] ⊂ O ∧ ∈ O opg ( g,ζ ) ξ opg ( g,ξ) . ( ) ( ) ¬ K ⊂ O ∈ O Set Q (ξ) = opg ( g,ξ) opg ( g,ξ) ,R(ζ) = ζ opg ( g,ζ ) , and P (ξ, ζ) = Q(ξ) ∧ R(ζ). Then, the above logical statement is equivalent to P (ξ, ζ) ∨ P(ζ, ξ). ∈ ¬ K { } ← But since λ opg ( g,λ) for every λ ξ, ζ , it consequently follows that R (λ) Q(λ) for every λ {ξ, ζ}. Therefore R (ξ) ∧ R(ζ) ← P(ξ, ζ). Because associativity with respect to ∧ holds, it then follows that P(ξ, ζ) ∨ P(ζ, ξ) → [R (ξ) ∧ R(ζ)] ∨ [R (ζ) ∧ R(ξ)] = R (ξ) ∧ R(ζ) . [ ] ∈ O O ∈ ∅ Hence, for every (ξ, ζ) g-FP [σ > 0], there exists ( g,ξ, g,ζ ) g-FO such that: [ ] [ ] ∈ O ∧ ∈ O ξ opg ( g,ξ) ζ opg ( g,ζ ) . ( ) T T (R) (R) T (R) This proves that, if Tg =( (Ω, g) is) a g- g -space g-Tg = Ω, g- g , then it is T (H) (H) T (H) Q.e.d. a g- g -space g-Tg = Ω, g- g . ( ) T T T (N) A necessary and( suﬃcient) condition for a g-space Tg = Ω, g to be a g- g - (N) T (N) P → space g-Tg = Ω, g- g may be given in terms of the g-operator opg : (Ω) P (Ω), a Tg-closed set Sg ⊂ Tg, Tg-open sets, and a Tg-closed set. ( ) Theorem . T T T (N) (N) ( )3.8 A g-space Tg = Ω, g is said to be a g- g -space g-Tg = T (N) Ω, g- g if and only if the following condition holds: ( ) ( ) ( ) [ ] S ⊂ Oˆ ⊂ ¬ Kˆ ⊂ O ∀S ∈ (3.5) g opg g,Sg opg g,Sg opg g,Sg g g-K Tg . Proof. T T (N) (N) S ∈ [ ] Necessity. Let the g-space Tg be a g- g -space g-Tg and, let g O ∈ T T g-K Tg and g,Sg g, respectively, be a g-Tg-closed set and a g-open neigh- S T (N) (N) bourhood set of g. Then, Tg is a g- g -space g-Tg implies that, for every (Kg,ξ, Kg,ζ ) ∈ g-F [∅], there exists (Og,ξ, Og,ζ ) ∈ g-F [∅] such that: [ K ] [ O ] O ⊃ ¬ K ∧ O ⊃ ¬ K opg ( g,ξ) opg ( g,ξ) opg ( g,ζ ) opg ( g,ζ ) . Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

THEORY OF g-Tg-SEPARATION AXIOMS 13 ( ) [ ] { O ∩¬ K ∅ ∈ S ∈ Clearly, opg ( g,ξ) opg ( g,ξ) = for any ξ Tg. The( relation) g g-K Tg implies that there exists a Kˆ ∈ ¬T such that S ⊇ ¬ op Kˆ and, O S ∈ T g,ξ g g g g,ξ ( g, g ) g is a T -open neighbourhood set of S implies that there exists Oˆ S , Kˆ S ∈ g ( g) ( ) g, g g, g T × ¬T such that O S ⊆ op Oˆ S ⊂ ¬ op Kˆ S . But, S ⊂ O S and, for g g g, g g g, (g ) g g(, g ) g g, g Oˆ ∈ ¬T ¬ Kˆ ⊂ Oˆ T (N) some g,ξ g, the relation opg g,Sg opg g,ξ holds in a g- g -space (N) g-Tg . Hence, ( ) ( ) ( ) ( ) ¬ Kˆ ⊆ S ⊂ Oˆ ⊂ ¬ Kˆ ⊂ Oˆ opg g,ξ g opg g,Sg opg g,Sg opg g,ξ [ ] S ∈ Oˆ ⊆ O for all g g-K Tg . At this stage, it suﬃces to set g,ξ g,Sg and the result follows. Suﬃciency. Conversely, suppose the following relation holds: ( ) ( ) ( ) [ ] S ⊂ Oˆ ⊂ ¬ Kˆ ⊂ O ∀S ∈ g opg g,Sg opg g,Sg opg g,Sg g g-K Tg . Then, its complementary reads ( ) ( ( )) ( ( )) ( ( )) { S ⊃ { Oˆ ⊃ { ¬ Kˆ ⊃ { O g opg g,Sg opg g,Sg opg g,Sg , ( ) ( ( )) [ ] ( ( )) { S { ¬ Kˆ S ∈ { Oˆ S where( ( g )), opg[ g], g g-O Tg are g-Tg-open sets( and,( opg)) g, g , { O S ∈ S ∩ { O S ∅ opg g, g [ ]g-K Tg are g-Tg-closed sets. Thus, g opg g, g = for S ∈ S ⊇ ¬ K K ∈ any g g-K Tg . But since the relation( ( g )) opg(( )g,ξ) holds( for( some)) g,ξ ¬T { O S ⊂ { S ⊆ { ¬ K g, it consequently( follows) that opg g,( g ) g opg g,ξ which, O ⊃ S ⊇ ¬ K K K ∈ in turn, implies opg g,Sg g opg g,ξ . Thus, for every ( g,ξ, g,ζ ) ∅ O O ∈ ∅ g-FK [ ], there exists ( g,ξ, g,ζ ) g-FO [ ] such that: [ ] [ ] O ⊃ ¬ K ∧ O ⊃ ¬ K opg ( g,ξ) opg ( g,ξ) opg ( g,ζ ) opg ( g,ζ ) . ( ) ( ) T T (N) (N) T (N) Therefore, Tg = Ω, g is a g- g -space g-Tg = Ω, g- g ; this completes the proof of the theorem. Q.e.d. ( ) Proposition . T T (N) (N) T (N) 3.9 If Tg =( (Ω, g) is) a g- g -space g-Tg = Ω, g- g , then it T (R) (R) T (R) is a g- g -space g-Tg = Ω, g- g . ( ) Proof. T T (N) (N) T (N) Let Tg = (Ω, g) be a g- g -space g-Tg = Ω, g- g . Then, for every K K ∈ ∅ O O ∈ ∅ ( g,ξ, g,ζ ) g-FK [ ], there exists ( g,ξ, g,ζ ) g-FO [ ] such that: [ ] [ ] O ⊃ ¬ K ∧ O ⊃ ¬ K opg ( g,ξ) opg ( g,ξ) opg ( g,ζ ) opg ( g,ζ ) . ( ) ( ) ¬ K ⊂ O ∈ O Set Q (ξ) = opg ( g,ξ) opg ( g,ξ) and R (ξ) = ξ opg ( g,ξ) so that the ∧ K K ∈ ∅ above logical statement now reads Q (ξ) Q(ζ). Then, since ( g,ξ, g,ζ ) g-FK [ ], Q(ξ) ∧ R(ζ) ← Q(ξ) and Q (ζ) ∧ R(ξ) ← Q(ζ) hold. Consequently, [ ] [ ] Q(ξ) ∧ R(ζ) ∧ Q(ζ) ∧ R(ξ) ← Q(ξ) ∧ Q(ζ) . But [ ] [ ] [ ] [ ] Q(ξ) ∧ R(ζ) ∨ Q(ζ) ∧ R(ξ) ← Q(ξ) ∧ R(ζ) ∧ Q(ζ) ∧ R(ξ) , and, therefore, [ ] [ ] Q(ξ) ∧ R(ζ) ∨ Q(ζ) ∧ R(ξ) ← Q(ξ) ∧ Q(ζ) . Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

14 KHODABOCUS M. I. AND SOOKIA N. U. H. [ ] ∈ K K ∈ ∅ ∈ Thus, for every (ξ, ζ) g-FP [σ > 0] and ( [g,ξ], g,ζ ) g-FK such that (ζ, ξ) / K K O O ∈ ∅ ( g,ξ, g,ζ ), there exists ( g,ξ, g,ζ ) g-FO such that: [( ) ( )] [( ¬ K ⊂ O ∧ ∈ O ∨ ¬ K opg ( g,ξ) opg ( g,ξ) ζ opg ( g,ζ ) opg ( g,ζ ) ) ( )] ⊂ O ∧ ∈ O opg ( g,ζ ) ξ opg ( g,ξ) . ( ) T T (N) (N) T (N) This proves that, if Tg =( (Ω, g) is) a g- g -space g-Tg = Ω, g- g , then it is T (R) (R) T (R) Q.e.d. a g- g -space g-Tg = Ω, g- g . T (H) T (F) By virtue of the above propositions, every g- g -space is a g- g -space, and T (K) T (N) T (R) hence, a g- g -space. Also, every g- g -space is a g- g -space, and hence, T (H) a g- g -space. But, the converse of both statements are untrue, and thus, the corollary follows. ⟨ ( )⟩ { } Corollary 3.10. If g-T(α) = Ω, g-T (α) , Λ = K, F, H, R, N , denotes a g g α∈Λ ( ) T (α) T T sequence of g- g -spaces, obtained⟨ after⟩ endowing a g-space Tg = Ω, g with the sequence of g-Tg,α-axioms g-Tg,α α∈Λ, then the following relations hold: (K) (F) (H) (R) (N) • i. Tg ⊆ Tg ⊆ Tg ⊆ Tg ⊆ Tg . • ii. ⇒ ⇒ ⇒ ⇒ g-Tg,N g-Tg,R g-Tg,H g-Tg,F g-Tg,K.

4. Discussion 4.1. Categorical Classifications. Having adopted{ a categorical} approach in ∈ T the classiﬁcations of the g-Tg,α-axioms, α Λ = K, F, H, R, N , in the g-space Tg, the aims here⟨ are, to⟩ establish the various relationships amongst the elements of the sequence g-Tg,α α∈Λ and, to illustrate them through diagrams. We have seen that, both the g-Tg,N, g-Tg,R-axioms imply the g-Tg,K, g-Tg,F- axioms and, on the other hand, the g-Tg,N-axiom implies the g-Tg,R-axiom and the g-Tg,H-axiom implies the g-Tg,F-axioms. The separation axioms diagram presented in Figs 1 illustrates these implications. ⟨ ⟩ We called the elements of the sequence -T -T -axioms. To this end ⟨ ⟩g g,α α∈Λ g g,α it does make sense to call those of T T -axioms. Thus, in a T -space ⟨ ⟩ g,α α∈Λ g,α g T = (Ω, T ), T stands for a sequence of separation axioms in the ordinary g g ⟨ g,α ⟩α∈Λ sense while g-T stands for its analogue but in the generalized sense, just g,α α∈Λ ⟨ ⟩ as, in a T -space = (Ω, T ), T stands for a sequence of separation axioms in T ⟨ α⟩ α∈Λ the ordinary sense while g-Tα α∈Λ stands for its analogue but in the generalized sense. Let FP [σ] = g-FP [σ] and set { ∩ } S def O O ∈ T × T O ⊆ S FO [ g] = ( g,ξ, g,ζ ) g g : λ=ξ,ζ g,λ g , { ∩ } S def K K ∈ ¬T × ¬T S ⊇ K (4.1) FK [ g] = ( g,ξ, g,ζ ) g g : g λ=ξ,ζ g,λ ,

where σ ≥ 0 and Sg ⊇ ∅. Then, the notions of Tg,K,Tg,F,Tg,H,Tg,R, and Tg,N- axioms in a Tg-space Tg = (Ω, Tg) may well be deﬁned as follows: • i. -Axiom ∈ O O ∈ T[ g,K ] : For every (ξ, ζ) FP [σ > 0], there exists ( g,ξ, g,ζ ) FO Sg ⊃ ∅ such that: [ ] [ ] (4.2) (ξ ∈ Og,ξ) ∧ (ζ∈ / Og,ξ) ∨ (ξ∈ / Og,ζ ) ∧ (ζ ∈ Og,ζ ) Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

THEORY OF g-Tg-SEPARATION AXIOMS 15

g-Tg,K

g-Tg,F g-Tg,H

g-T g-Tg,N g,R

Tg,R Tg,N

Tg,H Tg,F

Tg,K

Figure 1. Relationships: Separation axioms diagram.

• ii. -Axiom ∈ O O ∈ [Tg,F ] : For every (ξ, ζ) FP [σ > 0], there exists ( g,ξ, g,ζ ) FO Sg ⊃ ∅ such that: [ ] [ ] ∈ O ∧ ∈ O (4.3) (ξ, ζ) ×λ=ξ,ζ g,λ (ξ, ζ) / ×λ=ζ,ξ g,λ . • iii. -Axiom ∈ O O ∈ [T]g,H : For every (ξ, ζ) FP [σ > 0], there exists ( g,ξ, g,ζ ) FO ∅ such that: [ ] [ ] (4.4) ξ ∈ Og,ξ ∧ ζ ∈ Og,ζ . [ ] • iv. -Axiom ∈ K K ∈ ∅ Tg,R : For every (ξ, ζ) FP [σ > 0] and ( g,ξ, [ g],ζ ) FK such that (ζ, ξ) ∈/ (Kg,ξ, Kg,ζ ), there exists (Og,ξ, Og,ζ ) ∈ FO ∅ such that: [( ) ( )] [( ) ( )] (4.5) Kg,ξ ⊂ Og,ξ ∧ ζ ∈ Og,ζ ∨ Kg,ζ ⊂ Og,ζ ∧ ξ ∈ Og,ξ .

• v. Tg,N-Axiom: For every (Kg,ξ, Kg,ζ ) ∈ FK [∅], there exists (Og,ξ, Og,ζ ) ∈ FO [∅] such that: [ ] [ ] (4.6) Og,ξ ⊃ Kg,ξ ∧ Og,ζ ⊃ Kg,ζ . Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

16 KHODABOCUS M. I. AND SOOKIA N. U. H.

O ⊆ O K ⊇ ¬ K By virtue of the relations g,λ opg ( g,λ) and g,λ opg ( g,λ) for every O K ∈ T × ¬T × { } ← ( g,λ, g,λ, λ) g g ξ, ζ , these implications follow: g-Tg,K Tg,K, ← ← ← ← g-Tg,F Tg,F, g-Tg,H Tg,H, g-Tg,R Tg,R, and g-Tg,N Tg,N. When the statements preceding the above deﬁnitions are taken into account, another separation axioms diagram is obtained.⟨ In Fig.⟩ 2, we have⟨ illustrated⟩ the various relationships amongst the elements of g-Tg,α α∈Λ and Tg,α α∈Λ. It is interesting to present a third separation axioms diagram illustrating⟨ both⟩ the implications and ∈ 0 the categorical classiﬁcations of the elements of g-ν-Tg,α α∈Λ, where ν I3 .

g-Tg,N g-Tg,R g-Tg,H g-Tg,F g-Tg,K

Tg,N Tg,R Tg,H Tg,F Tg,K

Figure 2. Relationships: Separation axioms diagram.

∈ 0 ← For every fixed ν I3 , it is immediate that the implications g-ν-Tg,K ← ← ← g-ν-Tg,F, g-ν-Tg,F g-ν-Tg,H, g-ν-Tg,H g-ν-Tg,R, and g-ν-Tg,R g-ν-Tg,N hold. On the other hand, we saw in the first part of our works, on the theory of g-Tg-sets, that S ⊆ S ⊆ S ⊇ S ∀S ⊂ opg,0 ( g) opg,1 ( g) opg,3 ( g) opg,2 ( g) g Tg, ¬ S ⊇ ¬ S ⊇ ¬ S ⊆ ¬ S (4.7) opg,0 ( g) opg,1 ( g) opg,3 ( g) opg,2 ( g) , ¬ P → P as a consequence of our definitions of the g-operators opg,ν , opg,ν : (Ω) (Ω). ∈ → → ← Hence, for every α Λ, g-0-Tg,α g-1-Tg,α g-3-Tg,α and g-3-Tg,α g-2-Tg,α. When these properties are taken into consideration, the resulting separation axioms diagram so obtained is that presented in Fig. 3. It is reasonable to call them ∈ × 0 g-Tg,α-axioms of type α and of category ν, where (α, ν) Λ I3 . In order to exemplify the concept of g-Tg,α-axiom of type α and of category ν, ∈ × 0 where (α, ν) Λ I3 , a nice application is presented in the following section. 4.2.⟨ A Nice⟩ Application. Focusing{ on the fundamental} notions of the sequence g-T of g-T -axioms, Λ = K, F, H, R, N , in a T -space, founded upon g,α α∈Λ g,α g { T the class} of g- g-open sets, we shall now present a nice application. Let Ω = ξν : ν ∈ I∗ denotes the underlying set and consider the T -space T = (Ω, T ), where 3 { { } { } { } { } {g } { g } }g Tg (Ω) = ∅, ξ1 , ξ2 , ξ3 , ξ1, ξ2 , ξ1, ξ3 , ξ2, ξ3 , Ω { } (4.8) = Og,1, Og,2, Og,3, Og,4, Og,5, Og,6, Og,7, Og,8 , { { } { } { } { } { } { } ¬Tg (Ω) = Ω, ξ2, ξ3 , ξ1, ξ3 , ξ1, ξ2 , ξ3 , ξ2 , ξ1 , ∅} { } (4.9) = Kg,1, Kg,2, Kg,3, Kg,4, Kg,5, Kg,6, Kg,7, Kg,8 ,

respectively, stand for the classes of Tg-open and Tg-closed sets. In both settings, the T th O K ∈ ∗ g-open, closed sets occupying the ν position corresponds to g,ν , g,ν , ν I8 , T ∅ ∅ T O ⊆ O respectively, as is easily understood.(∪ Since) ∪ conditions g ( ) = , g ( g,ν ) g,ν ∈ ∗ T O T O for every ν I8 , and g ∈ ∗ g,ν = ∈ ∗ g ( g,ν ) are satisﬁed, it is clear ν I8 ({ν I8 }) T P → P ∈ ∗ that the one-valued map g : (Ω) ξν : ν I8 is a g-topology. After Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

THEORY OF g-Tg-SEPARATION AXIOMS 17

g-0-Tg,K g-1-Tg,K g-3-Tg,K g-2-Tg,K

g-0-Tg,F g-1-Tg,F g-3-Tg,F g-2-Tg,F

g-0-Tg,H g-1-Tg,H g-3-Tg,H g-3-Tg,H

g-0-Tg,R g-1-Tg,R g-3-Tg,R g-2-Tg,R

g-0-Tg,N g-1-Tg,N g-3-Tg,N g-2-Tg,N

Figure 3. Relationships: Separation axiom diagram.

{ } O ∈ ∗ × 0 T computing the elements of the set opg,ν ( g,ξα ):(α, ν) I8 I3 , called g- g- open sets, we obtain:

 {O O O O } ∀ ∈ { } × { }  g,2, g,5, g,6, g,8 (α, ν) 1 0, 2 ,   {O , O , O , O } ∀ (α, ν) ∈ {2} × {0, 2} ,  g,3 g,5 g,7 g,8  {Og,4, Og,6, Og,7, Og,8} ∀ (α, ν) ∈ {3} × {0, 2} , op (Og,ξ ) ∈ g,ν α  {K , K , K , K } ∀ (α, ν) ∈ {1} × {1, 3} ,  g,1 g,3 g,4 g,7   {Kg,1, Kg,2, Kg,4, Kg,6} ∀ (α, ν) ∈ {2} × {1, 3} ,  {Kg,1, Kg,2, Kg,3, Kg,5} ∀ (α, ν) ∈ {3} × {1, 3} . (4.10) Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

18 KHODABOCUS M. I. AND SOOKIA N. U. H. { } ¬ K ∈ ∗ × 0 T Similarly, the elements of opg,ν ( g,ξα ):(α, ν) I8 I3 , called g- g-closed sets, are:  {K K K K } ∀ ∈ { } × { }  g,1, g,3, g,4, g,7 (α, ν) 1 0, 2 ,   {K , K , K , K } ∀ (α, ν) ∈ {2} × {0, 2} ,  g,1 g,2 g,4 g,6  {Kg,1, Kg,2, Kg,3, Kg,5} ∀ (α, ν) ∈ {3} × {0, 2} , ¬ op (Kg,ξ ) ∈ g,ν α  {O , O , O , O } ∀ (α, ν) ∈ {1} × {1, 3} ,  g,2 g,5 g,6 g,8   {Og,3, Og,5, Og,7, Og,8} ∀ (α, ν) ∈ {2} × {1, 3} ,  {Og,4, Og,6, Og,7, Og,8} ∀ (α, ν) ∈ {3} × {1, 3} . (4.11) ∩ 0 op cl First, for every ν ∈ I , set I = op (Og,ξ ) and I = ∩ 3 g,(ξα,ξβ ) λ=α,β g,ν λ g,(ξα,ξβ ) ∗ ∗ 0 op ¬ op (K ,ξ ). Next, for all (α, β, ν) ∈ I × I × I , calculate I , λ=α,β g,ν g λ 3 3 3 g,(ξα,ξβ ) Icl . Finally, for every (r, s) ∈ I∗ × I∗, set O = (O , O ) and g,(ξα,ξβ ) 8 8 g,(r,s) g,r g,s Kg,(r,s) = (Kg,r, Kg,s). These procedures yield: ∪ { } ∈ ∗ \{ } g-FP [σ > 0] = (ξα, ξβ): β I3 α , α∈I∗ { 3 ∅ O O O O g-ν-FO [ ] = g,(3,2), g,(3,6), g,(4,2), g,(4,3), } Og,(4,5), Og,(6,3), Og,(7,2) , { ∅ K K K K g-ν-FK [ ] = g,(2,7), g,(3,6), g,(5,4), g,(5,6), } Kg,(5,7), Kg,(6,3), Kg,(6,7) , { } S ⊃ ∅ O ∈ ∗ × ∗ ⊃ ∅ ∀ ∈ 0 g-ν-FO [ g ] = g,(r,s) :(r, s) I8 I8 g-ν-FO [ ] ν I3 , { } S ⊃ ∅ K ∈ ∗ × ∗ ⊃ ∅ g-ν-FK [ g ] = g,(r,s) :(r, s) I8 I8 g-ν-FK [ ] . (4.12) { } We are now in a position to discuss the g-Tg,α-axioms, Λ = K, F, H, R, N . O ⊃ K O ⊃ K O ⊃ K O ∈ Let g,(p,q) g,(r,s) mean g,p g,r and g,q g,s, where( g,(p,q) ∅ K ∈ ∅ ∈ 0 O ⊃ g-ν-FO([ ] and ) g,(r,s) g-ν-F( K [ )]. Further, for( every) ν (I3 , let) opg,ν g,((p,q) ) ¬ K O ⊃ ¬ K O ⊃ ¬ K opg,ν g,(r,s) mean opg,ν g,p opg,ν g,r , opg,ν g,q opg,ν g,s . Then, the following relations are easily checked: Og,(7,2) ⊃ Kg,(2,7); Og,(6,3) ⊃ K O ⊃ K O ⊃ K O ⊃ K O ⊃ K g,(3,6); g,(4,5) g,(5,4); g,(4,3) g,(5,6); g,(4,2) g,(5,7); g,(3,6) ( g,(6,3)) O ⊃ K ∈ 0 O ⊆ O and g,(3,2) g,(6,7)(. But, for) every ν I3 , the relations g,(p,q) opg,ν g,(p,q) K ⊇ ¬ K and g,(r,s) opg,ν g,(r,s) hold for all (p, q) = (3, 2), (3, 6), (4, 2), (4, 3), (4, 5), (6, 3), (7, 2) and all (r, s) = (6, 7), (6, 3), (5, 7), (5, 6), (5, 4), (3, 6), (2, 7).( Combining) O ⊃ K O ⊃ these last( two) relations with g,(p,q) g,(r,s), it follows that opg,ν g,(p,q) ¬ K K ∈ ∅ O ∈ opg,ν g,(r,s) . Hence, for every g,(r,s) g-ν-FK [ ], there exists g,(p,q) g-ν-F [∅] such that: O [ ] [ ] O ⊃ ¬ K ∧ O ⊃ ¬ K opg,ν ( g,p) opg,ν ( g,r) opg,ν ( g,q) opg,ν ( g,s) . ( ) T (N) (N) T (N) This shows that Tg is a g- g -space g-Tg = Ω, g- g . [ ] ∈ K ∈ K ∈ K K ∈ ∅ Let (ξi, ξj) g,(r,s) mean ξi g,r and ξj g,s, where g,(r,s) g-ν-FK . Then, the following results are easily checked: (ξ2, ξ1) ∈ Kg,(2,7), Kg,(6,3), Kg,(6,7) Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

THEORY OF g-Tg-SEPARATION AXIOMS 19

and (ξ1, ξ2) ∈/ Kg,(2,7), Kg,(6,3), Kg,(6,7);(ξ3, ξ1) ∈ Kg,(2,7), Kg,(5,4), Kg,(5,7) and (ξ1, ξ3) ∈/ Kg,(2,7), Kg,(5,4), Kg,(5,7);(ξ3, ξ2) ∈ Kg,(3,6), Kg,(5,4), Kg,(5,6) and (ξ2, ξ3) ∈/ Kg,(3,6), Kg,(5,4), Kg,(5,6). But, Og,(7,2) ⊃ Kg,(2,7); Og,(6,3) ⊃ Kg,(3,6); Og,(4,5) ⊃ K O ⊃ K O ⊃ K O ⊃ K O ⊃ g,(5,4); g,(4,3) g,(5,6); g,(4,2) g,(5,7); g,(3,6) ( g,(6,3)) and g,(3,2) K ∈ 0 O ⊆ O g,(6,7). Furthermore, for every ν I3 , g,(p,q) opg,ν g,(p,q) (for all) (p, q) = K ⊇ ¬ K (3, 2), (3, 6), (4, 2), (4, 3), (4, 5), (6, 3), (7, 2) and g,(r,s) opg,ν g,(r,s) for all ∈ (r, s) = (2, 7), (3, 6), (5, 4), (5, 6), (5[ ,]7), (6, 3), (6, 7). Thus, for every (ξi, ξj) K ∈ ∅ ∈ K g-FP [σ > 0] and[ ]g,(r,s) g-ν-FK such that (ξj, ξi) / g,(r,s), there exists O ∈ ∅ g,(p,q) g-ν-FO such that: [( ) ( )] [( ¬ K ⊂ O ∧ ∈ O ∨ ¬ K opg,ν ( g,r) opg,ν ( g,p) ζ opg,ν ( g,q) opg,ν ( g,s) ) ( )] ⊂ O ∧ ∈ O opg,ν ( g,q) ξ opg,ν ( g,p) . ( ) T (R) (R) T (R) This shows that Tg is a g- g -space g-Tg = Ω, g- g . [ ] ∈ O ∈ O ∈ O O ∈ ∅ Let (ξi, ξj) g,(p,q) mean ξi g,p and ξj g,q, where g,(p,q) g-ν-FO . Then, the following relations are easily verified: (ξ2, ξ1) ∈ Og,(3,2), Og,(3,6), Og,(7,2) and (ξ1, ξ2) ∈/ Og,(3,2), Og,(3,6), Og,(7,2);(ξ3, ξ1) ∈ Og,(4,2), Og,(4,5), Og,(7,2) and ∈ O O O ∈ O O O ∈ (ξ1, ξ3) / g,(4,2), g,(4,5), g,(7,2);(ξ3, ξ2) g,(4,3), g,(4,5), g,(6,3)(and (ξ2,) ξ3) / O O O ∈ 0 O ⊆ O g,(4,3), g,(4,5), g,(6,3). But, for every ν I3 , g,(p,q) opg,ν g,(p,q) and O ∈ ∅ g,(p,q) g-ν-FO [ ] for all (p, q) = (3, 2), (3, 6), (4, 2), (4, 3), (4, 5),[ ] (6, 3), (7, 2). ∈ O ∈ ∅ Thus, for every (ξi, ξj) g-FP [σ > 0], there exists g,(p,q) g-ν-FO such that: [ ] [ ] ∈ O ∧ ∈ O ξi opg,ν ( g,p) ξj opg,ν ( g,q) . ( ) T (H) (H) T (H) This shows that Tg is a g- g -space g-Tg = Ω, g- g . ∈ O ∈ O ∈ O ∈ O Let (ξi, ξj) g[,(p,q) mean] ξi g,p, ξj g,q, and (ξj, ξi) / g,(p,q), where O ∈ S ⊃ ∅ g,(p,q) g-ν-FO g . Then, the following relations are easily verified: (ξ1, ξ2) ∈ Og,(p,q) and (ξ2, ξ1) ∈/ Og,(p,q) for all (p, q) = (2, 3), (2, 7), (6, 3), (6, 7); (ξ1, ξ3) ∈ Og,(p,q) and (ξ3, ξ1) ∈/ Og,(p,q) for all (p, q) = (2, 4), (2, 7), (5, 4), (5, 7); ∈ O ∈ O (ξ2, ξ3) g,(p,q) and( (ξ3, ξ1) )/ g,(p,q) for all (p, q) = (3, 4), (3, 6), (5, 4), (5, 6). O ⊆ O But, g,(p,q) opg,ν g,(p,q) for all (p, q) = (2, 3), (2, 4), (2, 7), (3, 4), (3, 6), ∈ (5, 4), (5, 6), (5, 7), (6,[3), (6, 7).] Hence, for every (ξi, ξj) g-FP [σ > 0], there O ∈ S ⊃ ∅ exists g,(p,q) g-ν-FO g such that: [ ] [ ] ∈ O ∧ ∈ O (ξi, ξj) ×λ=p,q opg,ν ( g,λ) (ξi, ξj) / ×λ=q,p opg,ν ( g,λ) . ( ) T (F) (F) T (F) This shows that Tg is a g- g -space g-Tg = Ω, g- g . ∈ O ∈ O ∈ O ∈ O ∈ O Let (ξi, ξj) g,(p,q) [mean ξi] g,p and ξj / g,p, or ξi / g,p and ξj g,q, O ∈ S ⊃ ∅ where g,(p,q) g-ν-FO g . Then, the following relations are easily verified: ∈ O O O O ∈ O O O (ξ1, ξ2) g,(2,3), g,(2,7), g,(6,3), g,(6,7);(ξ1, ξ3) g,(2,4), g,(2,7),( g,(5,4)), O ∈ O O O O O ⊆ O g,(5,7);(ξ2, ξ3) g,(3,4), g,(3,6), g,(5,4), g,(5,6). But, g,(p,q) opg,ν g,(p,q) for all (p, q) = (2, 3), (2, 4), (2, 7), (3, 4), (3, 6), (5, 4), (5, 6), (5, 7),[ (6, 3),] (6, 7). ∈ O ∈ S ⊃ ∅ Hence, for every (ξi, ξj) g-FP [σ > 0], there exists g,(p,q) g-ν-FO g such that: [( ( )) ( ( ))] [( ( )) ∈ O ∧ ∈ O ∨ ∈ O ξi opg,ν g,p ξj / opg,ν g,p ξi / opg,ν g,q ( ( ))] ∧ ∈ O ξj opg,ν g,q . ( ) T (K) (K) T (K) This shows that Tg is a g- g -space g-Tg = Ω, g- g . Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1

20 KHODABOCUS M. I. AND SOOKIA N. U. H.

The elements discussed in the preceding sections can be easily checked from this nice application. In the next section, we provide concluding remarks and future directions of the theory of g-Tg-separation axioms developed in the earlier sections. 4.3. Concluding Remarks. In this chapter, we developed a new theory, called Theory of g-Tg-Separation Axioms. The theory is based on the Theory of g-Tg-Sets but not on the Theory of g-Tg-Maps. In its own rights, the proposed theory has several advantages. The very first advantage is that the theory holds equally well when (Ω, Tg) = (Ω, T ) and other characteristics adapted on this ground, in which case it might be called Theory of g-T-Separation Axioms. T Thus, in a g-space the⟨ proposed⟩ theoretical framework categorises each element of the quintuple sequence g-T as g-T -axioms of type α and of categories g,α α∈{Λ g,α } ∈ × 0 ν, where (α, ν) Λ I3 and Λ = K, F, H, R, N and theorises the concepts in ⟨a unified⟩ way; in a T -space it categorises each element of the quintuple sequence g-T as g-T -axioms of type α and of categories ν, where (α, ν) ∈ Λ × I0 and {α α∈Λ α} 3 Λ = K, F, H, R, N and theorises the concepts in a unified way. Since the theory of g-Tg-separation Axioms has been based solely on theory of g-Tg-sets, as pointed out above, it is an interesting topic for future research either to develop the theory of g-Tg-separation axioms of mixed categories based on the afore- mentioned theory or to develop it but based on the theory of g-Tg-maps. More pre- ∈ 0 × 0 ̸ cisely, either for some pair (ν, µ) I3 I3 such that ν = µ, to develop the theory of {g-Tg-separation axioms based on the theory[ of] g-Tg-open[ ]} sets belonging to the class O O ∪O O O ∈ × g = g,ν g,µ :( g,ν , g,µ) {g-ν-O Tg g-µ-O Tg and the theory of[ g-T] g- K K ∪ K K K ∈ × closed[ sets]} belonging to the class g = g,ν g,µ :( g,ν , g,µ) g-ν-K Tg g-µ-K Tg in a Tg-space Tg or, to develop the theory of g-Tg-separation axioms based on the theory of g-Tg-maps, called g-(TΛ, TΘ)-continuous maps, g-({ TΛ, TΘ})- irresolute maps and g-(TΛ, TΘ)-homeomorphism maps, where Λ, Θ ∈ Ω, Σ, Υ , between any two of such Tg-spaces Tg,Ω, Tg,Σ, and Tg,Υ. Such two theories are what we thought would certainly be worth considering, and the discussion of this chapter ends here.

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Dr.Mohammad Irshad KHODABOCUS Current address: Department of Mathematics, Faculty of Science, University of Mauritius E-mail address: [email protected] Dr. Noor-Ul-Hacq SOOKIA Current address: Department of Mathematics, Faculty of Science, University of Mauritius E-mail address: [email protected]