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On Double Fuzzy Topological Spaces: Some New Types of Separation Axioms, Continuity, and Compactness

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17 April 2024

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17 April 2024

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Abstract
In this article, we first introduced new types of higher separation axioms called (r,s)-GFS-regular and (r,s)-GFS-normal spaces with the help of (r,s)-generalized fuzzy semi-closed sets (briefly, (r,s)-gfsc sets) and discussed some topological properties of them. Thereafter, we defined a stronger form of (r,s)-gfsc sets called (r,s)-g*fsc sets and investigated some of its features. Moreover, we showed that (r,s)-fsc set → (r,s)-g*fsc set → (r,s)-gfsc set, but the converse may not be true. In addition, we explored new types of fuzzy generalized mappings between double fuzzy topological spaces (U, τ, τ*) and (V, η, η*), and the relationships between these classes of mappings were examined with the help of some illustrative examples. Finally, some new types of compactness via (r,s)-gfso sets were defined and the relationships between them were introduced.
Keywords: 
Subject: Computer Science and Mathematics  -   Geometry and Topology

MSC:  03E72; 54A05; 54A40; 54C08; 54D15

1. Introduction and Preliminaries

The theory of fuzzy set was first presented by Zadeh [1]. Since then it has been improved and applied in most all the branches of technology and science, where theory of sets and mathematical logic play an important role. Also, many applications of these theory contributed to solving several practical problems in mathematics, social science, engineering, economics, etc. In recent years, many authors have contributed to fuzzy sets theory in the different directions in mathematics such as geometry, topology, algebra, operation research, see [2,3]. The concept of fuzzy sets was used to define fuzzy topological spaces in [4]. The study in [4] was particularly important in the development of the field of fuzzy topology, see e. g. [5–10]. The authors of [11–18] studied topological structures inspired by the hybridizations of soft sets [19] with fuzzy sets [1] and rough sets [20].
The concept of an intuitionistic fuzzy set was initiated by Atanassov [21,22], which is a generalization of a fuzzy set. Coker [23,24] introduced the concept of an intuitionistic fuzzy topological space based on the sense of Chang [4]. Later, Samanta and Mondal [25,26] gave the definition of an intuitionistic fuzzy topological space based on the sense of Šostak [27]. The name (intuitionistic) was replaced with the name (double) by Garcia and Rodabaugh [28]. The concept of ( r , s ) − g f c sets was introduced and investigated by Abbas [29]. Thereafter, the concept of ( r , s ) − s g f c sets was introduced by Zahran et al. [30] on double fuzzy topological space based on the sense of Šostak. Also, Taha [31] defined the concept of ( r , s ) − g f s c sets and some characterizations were given. So far, lots of spectacular and creative studies about the theories of an intuitionistic fuzzy set have been considered by some scholars, see e. g. [32–36].
The organization of this article is as follows:
• Firstly, we define new types of fuzzy separation axioms with the help of ( r , s ) − g f s c sets and establish some of their properties.
• Secondly, as a stronger form of ( r , s ) − g f s c sets [31], the notion of ( r , s ) − g ⊛ f s c sets is introduced and some properties are investigated. Moreover, we introduce new types of fuzzy mappings between double fuzzy topological spaces and relationships are obtained.
• Finally, some new types of compactness in double fuzzy topological spaces are defined and the relationships between them are specified.
• In the end, we give some conclusions and make a plan for future works in Section 5.
Throughout this article, nonempty sets will be denoted by V, U, etc. The family of all fuzzy sets on U is denoted by I U , and for μ ∈ I U , μ c ( u ) = 1 − μ ( u ) , for all u ∈ U (where I = [ 0 , 1 ] , I 1 = [ 0 , 1 ) , I ∘ = ( 0 , 1 ] ). Also, for t ∈ I , t ̲ ( u ) = t , for all u ∈ U .
A fuzzy point u t on U is a fuzzy set, defined as follows: u t ( k ) = t if k = u , and u t ( k ) = 0 for all k ∈ U − { u } . u t is said to belong to a fuzzy set μ , denoted by u t ∈ μ , if t ≤ μ ( u ) . The family of all fuzzy points on U is denoted by P t ( U ) .
A fuzzy set μ is a quasi-coincident with λ , denoted by μ q λ , if there is u ∈ U , such that μ ( u ) + λ ( u ) > 1 , if μ is not quasi-coincident with λ , we denote μ q ¯ λ .
The following results and notions will be used in the next sections:
Lemma 1.1.
[ 6 ] Let U be a nonempty set and ν , μ ∈ I U . Then,
(i) ν q μ iff there is u t ∈ ν such that u t q μ ,
(ii) ν ∧ μ ≠ 0 ̲ if ν q μ ,
(iii) ν q ¯ μ iff ν ≤ μ c ,
(iv) μ ≤ ν iff u t ∈ μ implies u t ∈ ν iff u t q μ implies u t q ν iff u t q ¯ ν implies u t q ¯ μ ,
(v) u t q ¯ ⋁ δ ∈ Δ ν δ iff there is δ 0 ∈ Δ such that u t q ¯ ν δ 0 .
Definition 1.1.
[ 25 , 30 ] A double fuzzy topology on U is a pair ( η , η * ) of the mappings η , η * : I U → I , which satisfy the following conditions.
(i) η ( ν ) + η * ( ν ) ≤ 1 , for each ν ∈ I U .
(ii) η ( ν 1 ∧ ν 2 ) ≥ η ( ν 1 ) ∧ η ( ν 2 ) and η * ( ν 1 ∧ ν 2 ) ≤ η * ( ν 1 ) ∨ η * ( ν 2 ) , for each ν 1 , ν 2 ∈ I U .
(iii) η ( ⋁ δ ∈ Δ ν δ ) ≥ ⋀ δ ∈ Δ η ( ν δ ) and η * ( ⋁ δ ∈ Δ ν δ ) ≤ ⋁ δ ∈ Δ η * ( ν δ ) , for each { ν δ } δ ∈ Δ ⊂ I U .
The triplet ( U , η , η * ) is said to be a double fuzzy topological space 〈briefly, dfts〉 in the sense of Šostak. η * ( ν ) and η ( ν ) may be interpreted as a gradation of nonopenness and a gradation of openness for ν ∈ I U , respectively.
In a dfts ( U , η , η * ) , the interior of ν ∈ I U , the closure of ν ∈ I U , the semi-closure of ν ∈ I U and the semi-interior of ν ∈ I U will be denoted by I η , η * ( ν , r , s ) , C η , η * ( ν , r , s ) , S C η , η * ( ν , r , s ) and S I η , η * ( ν , r , s ) , respectively [26,32,37].
Definition 1.2
(37,38). Let ( U , η , η * ) be a dfts, ν ∈ I U , r ∈ I ∘ , and s ∈ I 1 , then we have
(i) ν is said to be an ( r , s ) - f s c 〈resp., ( r , s ) - f p c and ( r , s ) - f r c 〉 set if ν ≥ I η , η * ( C η , η * ( ν , r , s ) , r , s ) 〈resp., ν ≥ C η , η * ( I η , η * ( ν , r , s ) , r , s ) and ν = C η , η * ( I η , η * ( ν , r , s ) , r , s ) 〉.
(ii) ν is said to be an ( r , s ) - f s o 〈resp., ( r , s ) - f p o and ( r , s ) - f r o 〉 set if ν ≤ C η , η * ( I η , η * ( ν , r , s ) , r , s ) 〈resp., ν ≤ I η , η * ( C η , η * ( ν , r , s ) , r , s ) and ν = I η , η * ( C η , η * ( ν , r , s ) , r , s ) 〉.
Definition 1.3
(29,30,31). Let ( U , η , η * ) be a dfts, μ , ν ∈ I U , r ∈ I ∘ , and s ∈ I 1 , then we have
(i) μ is said to be an ( r , s ) -generalized fuzzy closed 〈briefly, ( r , s ) - g f c 〉 set if C η , η * ( μ , r , s ) ≤ ν whenever μ ≤ ν and η ( ν ) ≥ r , η * ( ν ) ≤ s .
(ii) μ is said to be an ( r , s ) -semi generalized fuzzy closed 〈briefly, ( r , s ) - s g f c 〉 set if S C η , η * ( μ , r , s ) ≤ ν whenever μ ≤ ν and ν is ( r , s ) - f s o set.
(iii) μ is said to be an ( r , s ) -generalized fuzzy semi-closed 〈briefly, ( r , s ) - g f s c 〉 set if S C η , η * ( μ , r , s ) ≤ ν whenever μ ≤ ν and η ( ν ) ≥ r , η * ( ν ) ≤ s .
Definition 1.4.
[ 26 , 30 ] Let h : ( U , τ , τ * ) → ( V , η , η * ) be a mapping, then h is said to be
(i) DF -continuous if τ ( h − 1 ( λ ) ) ≥ η ( λ ) and τ * ( h − 1 ( λ ) ) ≤ η * ( λ ) for each λ ∈ I V .
(ii) DF -open if η ( h ( ν ) ) ≥ τ ( ν ) and η * ( h ( ν ) ) ≤ τ * ( ν ) for each ν ∈ I U .
(iii) DF -closed if η ( h c ( ν ) ) ≥ τ ( ν c ) and η * ( h c ( ν ) ) ≤ τ * ( μ c ) for each ν ∈ I U .
Definition 1.5.
[ 29 , 31 , 37 ] Let h : ( U , τ , τ * ) → ( V , η , η * ) be a mapping, r ∈ I ∘ , and s ∈ I 1 , then h is said to be
(i) DFS -continuous 〈resp., DFGS -continuous and DFG -continuous〉 if h − 1 ( μ ) is ( r , s ) - f s o 〈resp., ( r , s ) - g f s o and ( r , s ) - g f o 〉 set for each μ ∈ I V with η ( μ ) ≥ r , η * ( μ ) ≤ s .
(ii) DFGS -irresolute 〈resp., DF -irresolute〉 if h − 1 ( μ ) is ( r , s ) - g f s o 〈resp., ( r , s ) - f s o 〉 set for each μ ∈ I V is ( r , s ) - g f s o 〈resp., ( r , s ) - f s o 〉 set.
(iii) DFS -open 〈resp., DFGS -open and DFG -open〉 if h ( ν ) is ( r , s ) - f s o 〈resp., ( r , s ) - g f s o and ( r , s ) - g f o 〉 set for each ν ∈ I U with τ ( ν ) ≥ r , τ * ( ν ) ≤ s .
(iv) DFS -closed 〈resp., DFGS -closed and DFG -closed〉 if h ( ν ) is ( r , s ) - f s c 〈resp., ( r , s ) - g f s c and ( r , s ) - g f c 〉 set for each ν ∈ I U with τ ( ν c ) ≥ r , τ * ( ν c ) ≤ s .
The basic results and notions that we need in the next sections are found in [29–31,39–41].

2. Some New Higher Separation Axioms

Here, we are going to give the definitions of two types of higher fuzzy separation axioms with the help of ( r , s ) − g f s c sets [31] called ( r , s ) - GFS -regular 〈resp., ( r , s ) - GFS -normal〉 spaces and establish some of their properties.
Definition 2.1.
A dfts ( U , η , η * ) is said to be
(i) ( r , s ) - GFS -regular iff u t q ¯ μ for each μ ∈ I U is ( r , s ) − g f s c set implies that, there is ν δ ∈ I U with η ( ν δ ) ≥ r , η * ( ν δ ) ≤ s for δ ∈ { 1 , 2 } , such that u t ∈ ν 1 , μ ≤ ν 2 and ν 1 q ¯ ν 2 .
(ii) ( r , s ) - GFS -normal iff μ 1 q ¯ μ 2 for each ( r , s ) − g f s c sets μ δ ∈ I U for δ ∈ { 1 , 2 } implies that, there is ν δ ∈ I U with η ( ν δ ) ≥ r and η * ( ν δ ) ≤ s , such that μ δ ≤ ν δ and ν 1 q ¯ ν 2 .
Theorem 2.1.
Let ( U , η , η * ) be a d f t s , r ∈ I ∘ , and s ∈ I 1 , then the following statements are equivalent.
(i) ( U , η , η * ) is ( r , s ) - GFS -regular space.
(ii) If u t ∈ λ for each λ ∈ I U is ( r , s ) − g f s o , there is μ ∈ I U with η ( μ ) ≥ r and η * ( μ ) ≤ s , such that u t ∈ μ ≤ C η , η * ( μ , r , s ) ≤ λ .
(iii) If u t q ¯ λ for each λ ∈ I U is ( r , s ) − g f s c , there is μ δ ∈ I U with η ( μ δ ) ≥ r , η * ( μ δ ) ≤ s for δ ∈ { 1 , 2 } , such that u t ∈ μ 1 , λ ≤ μ 2 and C η , η * ( μ 1 , r , s ) q ¯ C η , η * ( μ 2 , r , s ) .
Proof. (i) ⇒ (ii) Let u t ∈ λ for each λ ∈ I U is an ( r , s ) − g f s o , then u t q ¯ λ c for ( r , s ) − g f s c set λ c . Since ( U , η , η * ) is ( r , s ) - GFS -regular, there is μ , ν ∈ I U with η ( μ ) ≥ r , η * ( μ ) ≤ s and η ( ν ) ≥ r , η * ( ν ) ≤ s such that u t ∈ μ , λ c ≤ ν and μ q ¯ ν . It implies u t ∈ μ ≤ ν c ≤ λ . Since η ( ν ) ≥ r and η * ( ν ) ≤ s , u t ∈ μ ≤ C η , η * ( μ , r , s ) ≤ λ .
(ii) ⇒ (iii) Let u t q ¯ λ for each λ ∈ I U is an ( r , s ) − g f s c , then u t ∈ λ c for ( r , s ) − g f s o set λ c . By (ii), there is μ ∈ I U with η ( μ ) ≥ r , η * ( μ ) ≤ s such that u t ∈ μ ≤ C η , η * ( μ , r , s ) ≤ λ c . Since η ( μ ) ≥ r and η * ( μ ) ≤ s , then μ is ( r , s ) − g f s o and u t ∈ μ . Again, by (ii), there is μ 1 ∈ I U with η ( μ 1 ) ≥ r , η * ( μ 1 ) ≤ s such that
u t ∈ μ 1 ≤ C η , η * ( μ 1 , r , s ) ≤ μ ≤ C η , η * ( μ , r , s ) ≤ λ c .
It implies λ ≤ ( C η , η * ( μ , r , s ) ) c = I η , η * ( μ c , r , s ) ≤ μ c . Put μ 2 = I η , η * ( μ c , r , s ) , then η ( μ 2 ) ≥ r , η * ( μ 2 ) ≤ s .
So, C η , η * ( μ 2 , r , s ) ≤ μ c ≤ ( C η , η * ( μ 1 , r , s ) ) c , that is, C η , η * ( μ 1 , r , s ) q ¯ C η , η * ( μ 2 , r , s ) .
(iii) ⇒ (i) It is trivial. □
In a similar way, we can prove the following corollary.
Corollary 2.1.
Let ( U , η , η * ) be a d f t s , r ∈ I ∘ , and s ∈ I 1 , then the following statements are equivalent.
(i) ( U , η , η * ) is ( r , s ) - GFS -normal space.
(ii) If ν ≤ λ for each ν ∈ I U is ( r , s ) − g f s c and λ ∈ I U is ( r , s ) − g f s o set, there is μ ∈ I U with η ( μ ) ≥ r and η * ( μ ) ≤ s , such that ν ≤ μ ≤ C η , η * ( μ , r , s ) ≤ λ .
(iii) If λ 1 q ¯ λ 2 for each ( r , s ) − g f s c sets λ δ ∈ I U for δ ∈ { 1 , 2 } , there is μ δ ∈ I U with η ( μ δ ) ≥ r and η * ( μ δ ) ≤ s , such that λ δ ≤ μ δ and C η , η * ( μ 1 , r , s ) q ¯ C η , η * ( μ 2 , r , s ) .
Theorem 2.2.
If h : ( U , τ , τ * ) → ( V , η , η * ) is DF -irresolute, DF -open and bijective map, and ( U , τ , τ * ) is ( r , s ) - GFS -regular 〈resp., ( r , s ) - GFS -normal〉 space, then ( V , η , η * ) is ( r , s ) - GFS -regular 〈resp., ( r , s ) - GFS -normal〉 space.
Proof. 
Let v t q ¯ μ for each μ ∈ I V is ( r , s ) − g f s c . Since h is DF -irresolute, DF -open and bijective map, then by Theorem 4.11 [31], h is DFGS -irresolute. Hence, h − 1 ( μ ) is ( r , s ) − g f s c set. Put v t = h ( u t ) . Then, u t q ¯ h − 1 ( μ ) . Since ( U , τ , τ * ) is ( r , s ) - GFS -regular, there is μ δ ∈ I U with τ ( μ δ ) ≥ r , τ * ( μ δ ) ≤ s and δ ∈ { 1 , 2 } such that u t ∈ μ 1 , h − 1 ( μ ) ≤ μ 2 and μ 1 q ¯ μ 2 . Since h is DF -open and bijective map, we have
v t ∈ h ( μ 1 ) , μ = h ( h − 1 ( μ ) ) ≤ h ( μ 2 ) , h ( μ 1 ) q ¯ h ( μ 2 ) .
Hence, ( V , η , η * ) is ( r , s ) - GFS -regular space. The other case follows similar lines. □
Theorem 2.3.
If h : ( U , τ , τ * ) → ( V , η , η * ) is DF -continuous, DFGS -irresolute closed and injective map, and ( V , η , η * ) is ( r , s ) - GFS -regular 〈resp., ( r , s ) - GFS -normal〉, then ( U , τ , τ * ) is ( r , s ) - GFS -regular 〈resp., ( r , s ) - GFS -normal〉.
Proof. 
Let u t q ¯ λ for each λ ∈ I U is ( r , s ) − g f s c . Since h is DFGS -irresolute closed, h ( λ ) is ( r , s ) − g f s c . Since h is injective, u t q ¯ λ implies h ( u t ) q ¯ h ( λ ) . Since ( V , η , η * ) is ( r , s ) - GFS -regular, there is μ δ ∈ I U with η ( μ δ ) ≥ r , η * ( μ δ ) ≤ s and δ ∈ { 1 , 2 } such that h ( u t ) ∈ μ 1 , h ( λ ) ≤ μ 2 and μ 1 q ¯ μ 2 . Since h is DF -continuous, u t ∈ h − 1 ( μ 1 ) , λ ≤ h − 1 ( μ 2 ) with η ( h − 1 ( μ δ ) ) ≥ r , η * ( h − 1 ( μ δ ) ) ≤ s and δ ∈ { 1 , 2 } and h − 1 ( μ 1 ) q ¯ h − 1 ( μ 2 ) . Hence, ( U , τ , τ * ) is ( r , s ) - GFS -regular. The other case follows similar lines. □
Theorem 2.4.
If h : ( U , τ , τ * ) → ( V , η , η * ) is DFGS -irresolute, DF -open, DF -closed and surjective map, and ( U , τ , τ * ) is ( r , s ) - GFS -regular 〈resp., ( r , s ) - GFS -normal〉, then ( V , η , η * ) is ( r , s ) - GFS -regular 〈resp., ( r , s ) - GFS -normal〉.
Proof. 
Let v t ∈ μ for each μ ∈ I V is ( r , s ) − g f s o . Since h is DFGS -irresolute and surjective then, there is u ∈ h − 1 ( { v } ) such that u t ∈ h − 1 ( μ ) with ( r , s ) − g f s o set h − 1 ( μ ) . Since ( U , τ , τ * ) is ( r , s ) - GFS -regular, by Theorem 2.1, there is ν ∈ I U with τ ( ν ) ≥ r , τ * ( ν ) ≤ s such that u t ∈ ν ≤ C τ , τ * ( ν , r , s ) ≤ h − 1 ( μ ) . It implies
v t ∈ h ( ν ) ≤ h ( C τ , τ * ( ν , r , s ) ) ≤ μ .
Since h is DF -open and DF -closed, then η ( h ( ν ) ) ≥ r , η * ( h ( ν ) ) ≤ s and η ( h c ( C τ , τ * ( ν , r , s ) ) ) ≥ r . Hence, v t ∈ h ( ν ) ≤ C η , η * ( h ( ν ) , r , s ) ≤ C η , η * ( h ( C τ , τ * ( ν , r , s ) ) , r , s ) ≤ μ . Thus, ( V , η , η * ) is ( r , s ) - GFS -regular. The other case follows similar lines. □

3. A Stronger Novel form of ( r , s ) − gfsc Sets

Here, we introduce and study a stronger form of ( r , s ) -generalized fuzzy semi-closed sets 〈briefly, ( r , s ) − g f s c 〉 called ( r , s ) − g ⊛ f s c sets. Also, we show that ( r , s ) − f s c set [37] ⇒ ( r , s ) − g ⊛ f s c set ⇒ ( r , s ) − g f s c set [31], but the converse may not be true. After that, we introduce new types of fuzzy mappings between double fuzzy topological spaces and relationships are obtained.
Definition 3.1.
Let ( V , η , η * ) be a d f t s , ν , ρ ∈ I V , r ∈ I ∘ , and s ∈ I 1 , then we have:
(i) A fuzzy set ρ is said to be an ( r , s ) -strongly generalized fuzzy semi-closed 〈briefly, ( r , s ) − g ⊝ f s c 〉 if S C η , η * ( ρ , r , s ) ≤ ν whenever ρ ≤ ν and ν is ( r , s ) − g f o set,
(ii) A fuzzy set ρ is said to be an ( r , s ) -strongly* generalized fuzzy semi-closed 〈briefly, ( r , s ) − g ⊛ f s c 〉 if S C η , η * ( ρ , r , s ) ≤ ν whenever ρ ≤ ν and ν is ( r , s ) − g f s o set.
Remark 3.1.
(i) A fuzzy set ρ ∈ I V is ( r , s ) − g ⊝ f s o if ρ c is ( r , s ) − g ⊝ f s c set.
(ii) A fuzzy set ρ ∈ I V is ( r , s ) − g ⊛ f s o if ρ c is ( r , s ) − g ⊛ f s c set.
Remark 3.2.
From the previous definition, we can summarize the relationships among different types of fuzzy soft sets as in the next diagram.
( r , s ) − f s c → ( r , s ) − g ⊛ f s c
↓ ↓
( r , s ) − s g f c ( r , s ) − g ⊝ f s c
↓ ↓
( r , s ) − g f s c
Remark 3.3.
The converses of the above implications may not be true, as shown by Examples 3.1, 3.2, 3.3 and 3.4.
Example 3.1.
Let V = { v 1 , v 2 , v 3 , v 4 } and ρ , ν ∈ I V defined as follows: ρ = { v 1 1 . 0 , v 2 1 . 0 , v 3 1 . 0 , v 4 0 . 0 } and ν = { v 1 0 . 0 , v 2 0 . 0 , v 3 1 . 0 , v 4 1 . 0 } . Also, ( η , η * ) defined on V as follows:
η ( μ ) = 1 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 2 , if μ = ν , 0 , otherwise ,
η * ( μ ) = 0 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 2 , if μ = ν , 1 , otherwise .
Thus, ρ is ( 1 2 , 1 2 ) − g ⊛ f s c set, but it is not ( 1 2 , 1 2 ) − f s c set.
Example 3.2.
Let V = { v 1 , v 2 , v 3 , v 4 } and ρ , λ 1 , λ 2 , λ 3 ∈ I V defined as follows: ρ = { v 1 1 . 0 , v 2 0 . 0 , v 3 1 . 0 , v 4 0 . 0 } , λ 1 = { v 1 0 . 0 , v 2 1 . 0 , v 3 1 . 0 , v 4 1 . 0 } , λ 2 = { v 1 0 . 0 , v 2 1 . 0 , v 3 1 . 0 , v 4 0 . 0 } and λ 3 = { v 1 0 . 0 , v 2 0 . 0 , v 3 1 . 0 , v 4 0 . 0 } . Also, ( η , η * ) defined on V as follows:
η ( μ ) = 1 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 4 , if μ ∈ { λ 1 , λ 2 , λ 3 } , 0 , otherwise ,
η * ( μ ) = 0 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 4 , if μ ∈ { λ 1 , λ 2 , λ 3 } , 1 , otherwise .
Thus, ρ is ( 1 4 , 1 4 ) − g ⊝ f s c set, but it is not ( 1 4 , 1 4 ) − g ⊛ f s c set.
Example 3.3.
Let V = { v 1 , v 2 , v 3 } and ν , μ 1 , μ 2 ∈ I V defined as follows: ν = { v 1 1 . 0 , v 2 0 . 0 , v 3 0 . 0 } , μ 1 = { v 1 0 . 0 , v 2 0 . 0 , v 3 1 . 0 } and μ 2 = { v 1 1 . 0 , v 2 1 . 0 , v 3 0 . 0 } . Also, ( η , η * ) defined on V as follows:
η ( μ ) = 1 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 2 , if μ ∈ { μ 1 , μ 2 } , 0 , otherwise ,
η * ( μ ) = 0 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 2 , if μ ∈ { μ 1 , μ 2 } , 1 , otherwise .
Thus, ν is ( 1 2 , 1 2 ) − s g f c set, but it is not ( 1 2 , 1 2 ) − g ⊛ f s c set.
Example 3.4.
Let V = { v 1 , v 2 , v 3 } and ν , μ 1 , μ 2 ∈ I V defined as follows: ν = { v 1 1 . 0 , v 2 0 . 0 , v 3 1 . 0 } , μ 1 = { v 1 1 . 0 , v 2 0 . 0 , v 3 0 . 0 } and μ 2 = { v 1 1 . 0 , v 2 1 . 0 , v 3 0 . 0 } . Also, ( η , η * ) defined on V as follows:
η ( μ ) = 1 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 3 , if μ ∈ { μ 1 , μ 2 } , 0 , otherwise ,
η * ( μ ) = 0 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 3 , if μ ∈ { μ 1 , μ 2 } , 1 , otherwise .
Thus, ν is ( 1 3 , 1 3 ) − g f s c set, but it is not ( 1 3 , 1 3 ) − g ⊛ f s c set.
Remark 3.4.
In general, ( r , s ) − g f c sets [29] and ( r , s ) − g ⊛ f s c sets are independent concepts, as shown by Example 3.5.
Example 3.5.
Let V = { v 1 , v 2 , v 3 , v 4 } and ρ , ν , μ 1 , μ 2 ∈ I V defined as follows: ρ = { v 1 0 . 0 , v 2 1 . 0 , v 3 0 . 0 , v 4 0 . 0 } , ν = { v 1 1 . 0 , v 2 1 . 0 , v 3 0 . 0 , v 4 1 . 0 } , μ 1 = { v 1 1 . 0 , v 2 0 . 0 , v 3 0 . 0 , v 4 0 . 0 } and μ 2 = { v 1 1 . 0 , v 2 1 . 0 , v 3 0 . 0 , v 4 0 . 0 } . Also, ( η , η * ) defined on V as follows:
η ( μ ) = 1 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 2 , if μ ∈ { μ 1 , μ 2 } , 0 , otherwise ,
η * ( μ ) = 0 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 2 , if μ ∈ { μ 1 , μ 2 } , 1 , otherwise .
Thus, ρ is ( 1 2 , 1 2 ) − g ⊛ f s c set, but it is not ( 1 2 , 1 2 ) − g f c set. Also, ν is ( 1 2 , 1 2 ) − g f c set, but it is not ( 1 2 , 1 2 ) − g ⊛ f s c set.
Remark 3.5.
In general, any intersection of ( r , s ) − g ⊛ f s o sets is not ( r , s ) − g ⊛ f s o , and any union of ( r , s ) − g ⊛ f s c sets is not ( r , s ) − g ⊛ f s c , as shown by Example 3.6.
Example 3.6.
Let V = { v 1 , v 2 , v 3 , v 4 } and ν , ρ , μ 1 , μ 2 , μ 3 ∈ I V defined as follows: ν = { v 1 1 . 0 , v 2 0 . 0 , v 3 1 . 0 , v 4 1 . 0 } , ρ = { v 1 0 . 0 , v 2 1 . 0 , v 3 1 . 0 , v 4 1 . 0 } , μ 1 = { v 1 1 . 0 , v 2 0 . 0 , v 3 0 . 0 , v 4 0 . 0 } , μ 2 = { v 1 0 . 0 , v 2 1 . 0 , v 3 0 . 0 , v 4 0 . 0 } and μ 3 = { v 1 1 . 0 , v 2 1 . 0 , v 3 0 . 0 , v 4 0 . 0 } . Also, ( η , η * ) defined on V as follows:
η ( μ ) = 1 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 3 , if μ ∈ { μ 1 , μ 2 , μ 3 } , 0 , otherwise ,
η * ( μ ) = 0 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 3 , if μ ∈ { μ 1 , μ 2 , μ 3 } , 1 , otherwise .
Thus, μ 1 and μ 2 are ( 1 3 , 1 3 ) − g ⊛ f s c sets, but μ 1 ∨ μ 2 is not ( 1 3 , 1 3 ) − g ⊛ f s c . Also, ρ and ν are ( 1 3 , 1 3 ) − g ⊛ f s o sets, but ρ ∧ ν is not ( 1 3 , 1 3 ) − g ⊛ f s o .
Theorem 3.1.
Let ( V , η , η * ) be a d f t s , μ , λ ∈ I V , r ∈ I ∘ , and s ∈ I 1 , then λ is ( r , s ) − g ⊛ f s c set iff every μ is ( r , s ) − g f s o set and λ ≤ μ , there is ρ is ( r , s ) − f s c set, such that λ ≤ ρ ≤ μ .
Proof. 
(⇒) Let λ be an ( r , s ) − g ⊛ f s c , λ ≤ μ and μ be an ( r , s ) − g f s o set, then S C η , η * ( λ , r , s ) ≤ μ . Put ρ = S C η , η * ( λ , r , s ) , there is ρ is ( r , s ) − f s c set such that λ ≤ ρ ≤ μ .
(⇐) Assume that λ ≤ μ and μ is ( r , s ) − g f s o set, then by hypothesis, there is ρ is ( r , s ) − f s c set such that λ ≤ ρ ≤ μ , therefore, S C η , η * ( λ , r , s ) ≤ μ . So, λ is ( r , s ) − g ⊛ f s c set. □
Proposition 3.1.
Let ( V , η , η * ) be a d f t s , μ , λ ∈ I V , r ∈ I ∘ , and s ∈ I 1 , then the following properties hold.
(i) If λ is ( r , s ) − g ⊛ f s c and λ ≤ μ ≤ S C η , η * ( λ , r , s ) , then μ is ( r , s ) − g ⊛ f s c set.
(ii) If λ is ( r , s ) − g ⊛ f s o and S I η , η * ( λ , r , s ) ≤ μ ≤ λ , then μ is ( r , s ) − g ⊛ f s o set.
(iii) If one of the following two cases hold:
(a) λ is ( r , s ) − g ⊛ f s c and ( r , s ) − g f s o .
(b) λ is ( r , s ) − g ⊛ f s c and η ( λ ) ≥ r , η * ( λ ) ≤ s .
Then, λ is ( r , s ) − f s c set.
Proof. 
(i) Let ν be an ( r , s ) − g f s o set and μ ≤ ν , then λ ≤ ν . Since λ is ( r , s ) − g ⊛ f s c set, hence S C η , η * ( λ , r , s ) ≤ ν , but μ ≤ S C η , η * ( λ , r , s ) . Then, S C η , η * ( μ , r , s ) ≤ ν . So, μ is ( r , s ) − g ⊛ f s c set.
(ii) and (iii) are easily proved by a similar way. □
Theorem 3.2.
Let ( V , η , η * ) be a d f t s , ν ∈ I V , s ∈ I 1 , and r ∈ I ∘ , then the following statements are equivalent.
(i) ν is ( r , s ) − f r o set.
(ii) ν is ( r , s ) − g ⊛ f s c set and η ( ν ) ≥ r , η * ( ν ) ≤ s .
Proof. 
(i) ⇒ (ii) Let μ ∈ I V be an ( r , s ) − g f s o set and ν ≤ μ . Since ν is ( r , s ) − f r o set, then ν ∨ I η , η * ( C η , η * ( ν , r , s ) , r , s ) = ν ≤ μ . So, S C η , η * ( ν , r , s ) ≤ μ , and hence ν is ( r , s ) − g ⊛ f s c set.
(ii) ⇒ (i) Since ν is ( r , s ) − g ⊛ f s c set and η ( ν ) ≥ r , η * ( ν ) ≤ s , then by Proposition 3.1(iii), ν is ( r , s ) − f s c set. But, ν is ( r , s ) − f p o set. Therefore, ν is ( r , s ) − f r o set. □
Theorem 3.3.
Let ( V , η , η * ) be a d f t s , ρ , μ , ν ∈ I V , s ∈ I 1 , and r ∈ I ∘ , then the following statements are equivalent.
(i) ν is ( r , s ) − g ⊛ f s o set.
(ii) For any μ is ( r , s ) − g f s c set and μ ≤ ν , then μ ≤ S I η , η * ( ν , r , s ) .
(iii) For any μ is ( r , s ) − g f s c set and μ ≤ ν , there is ρ is ( r , s ) − f s o set such that μ ≤ ρ ≤ ν .
Proof. 
(i) ⇒ (ii) Let μ be an ( r , s ) − g f s c set and μ ≤ ν . Then, ν c ≤ μ c , which is ( r , s ) − g f s o set. Hence, S C η , η * ( ν c , r , s ) ≤ μ c implies μ ≤ ( S C η , η * ( ν c , r , s ) ) c . Then, μ ≤ S I η , η * ( ν , r , s ) .
(ii) ⇒ (iii) Let μ be an ( r , s ) − g f s c set and μ ≤ ν . Then, by hypothesis μ ≤ S I η , η * ( ν , r , s ) . Put S I η , η * ( ν , r , s ) = ρ . Hence, μ ≤ ρ ≤ ν .
(iii) ⇒ (i) Let μ be an ( r , s ) − g f s o set and ν c ≤ μ . Then, μ c ≤ ν and by hypothesis, there is ρ is ( r , s ) − f s o set such that μ c ≤ ρ ≤ ν , that is, ν c ≤ ρ c ≤ μ . Therefore, by Theorem 3.1, ν c is ( r , s ) − g ⊛ f s c set. Hence, ν is ( r , s ) − g ⊛ f s o set. □
Definition 3.2.
Let h : ( U , τ , τ * ) → ( V , η , η * ) be a mapping, then h is said to be
(i) Strongly* double fuzzy generalized semi-continuous 〈briefly, S * DFGS -continuous〉 if h − 1 ( ν ) is ( r , s ) − g ⊛ f s o set for each ν ∈ I V and η ( ν ) ≥ r , η * ( ν ) ≤ s .
(ii) S * DFGS -irresolute if h − 1 ( ν ) is ( r , s ) − g ⊛ f s o set for each ν ∈ I V is ( r , s ) − g ⊛ f s o set.
(iii) S * DFGS -open if h ( ρ ) is ( r , s ) − g ⊛ f s o set for each ρ ∈ I U and τ ( ρ ) ≥ r , τ * ( ρ ) ≤ s .
(iv) S * DFGS -closed if h ( ρ ) is ( r , s ) − g ⊛ f s c set for ρ ∈ I U and τ ( ρ c ) ≥ r , τ * ( ρ c ) ≤ s .
Remark 3.6.
From the previous definitions, we can summarize the relationships among different types of DF -continuity as in the next diagram.
DF − c o n t i n u i t y
↙ ↘
DFG − c o n t i n u i t y DFS − c o n t i n u i t y
↓ ↓
DFGS − c o n t i n u i t y ⟵ S * DFGS − c o n t i n u i t y
Remark 3.7.
The converses of the above implications may not be true, as shown by Examples 3.7 and 3.8.
Example 3.7.
Let V = { v 1 , v 2 , v 3 , v 4 } and ρ , ν ∈ I V defined as follows: ρ = { v 1 0 . 0 , v 2 0 . 0 , v 3 1 . 0 , v 4 1 . 0 } and ν = { v 1 0 . 0 , v 2 0 . 0 , v 3 0 . 0 , v 4 1 . 0 } . Define η , η * , τ , τ * : I V ⟶ I as follows:
η ( μ ) = 1 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 2 , if μ = ρ , 0 , otherwise ,
η * ( μ ) = 0 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 2 , if μ = ρ , 1 , otherwise ,
τ ( μ ) = 1 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 2 , if μ = ν , 0 , otherwise ,
τ * ( μ ) = 0 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 2 , if μ = ν , 1 , otherwise .
Thus, the identity mapping i d v : ( V , η , η * ) → ( V , τ , τ * ) is S * DFGS -continuous, but it is not DFS -continuous.
Example 3.8.
Let V = { v 1 , v 2 , v 3 } and μ 1 , μ 2 , μ 3 ∈ I V defined as follows: μ 1 = { v 1 0 . 0 , v 2 0 . 0 , v 3 1 . 0 } , μ 2 = { v 1 1 . 0 , v 2 1 . 0 , v 3 0 . 0 } and μ 3 = { v 1 0 . 0 , v 2 1 . 0 , v 3 1 . 0 } . Define η , η * , τ , τ * : I V ⟶ I as follows:
η ( μ ) = 1 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 2 , if μ ∈ { μ 1 , μ 2 } , 0 , otherwise ,
η * ( μ ) = 0 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 2 , if μ ∈ { μ 1 , μ 2 } , 1 , otherwise ,
τ ( μ ) = 1 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 2 , if μ = μ 3 , 0 , otherwise ,
τ * ( μ ) = 0 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 2 , if μ = μ 3 , 1 , otherwise .
Thus, the identity mapping i d v : ( V , η , η * ) → ( V , τ , τ * ) is DFGS -continuous, but it is not S* DFGS -continuous.
Lemma 3.1.
Every S * DFGS -irresolute mapping is S * DFGS -continuous.
Remark 3.8.
The converse of Lemma 3.1 may not be true, as shown by Example 3.9.
Example 3.9.
Let V = { v 1 , v 2 } . Define η , η * , τ , τ * : I V ⟶ I as follows:
η ( ρ ) = 1 , if ρ ∈ { 0 ̲ , 1 ̲ } , 1 2 , if ρ ∈ { 0.1 ̲ , 0.3 ̲ } , 0 , otherwise ,
η * ( ρ ) = 0 , if ρ ∈ { 0 ̲ , 1 ̲ } , 1 2 , if ρ ∈ { 0.1 ̲ , 0.3 ̲ } , 1 , otherwise ,
τ ( ρ ) = 1 , if ρ ∈ { 0 ̲ , 1 ̲ } , 1 2 , if ρ = 0.1 ̲ , 0 , otherwise ,
τ * ( ρ ) = 0 , if ρ ∈ { 0 ̲ , 1 ̲ } , 1 2 , if ρ = 0.1 ̲ , 1 , otherwise .
Thus, the identity mapping i d v : ( V , η , η * ) → ( V , τ , τ * ) is S * DFGS -continuous, but it is not S * DFGS -irresolute.

4. New Types of Compactness

Here, several types of compactness in double fuzzy topological spaces were introduced and the relationships between them were studied.
Definition 4.1.
Let ( U , η , η * ) be a d f t s , r ∈ I ∘ , and s ∈ I 1 , then μ ∈ I U is said to be an ( r , s ) -fuzzy compact iff for each family { λ j ∈ I U | η ( λ j ) ≥ r and η * ( λ j ) ≤ s } j ∈ ϝ , such that μ ≤ ⋁ j ∈ ϝ λ j , there is a finite subset ϝ ∘ of ϝ , such that μ ≤ ⋁ j ∈ ϝ ∘ λ j .
Definition 4.2.
Let ( U , η , η * ) be a d f t s , r ∈ I ∘ , and s ∈ I 1 , then μ ∈ I U is said to be an ( r , s ) -fuzzy GS -compact iff for each family { λ j ∈ I U | λ j is ( r , s ) − g f s o } j ∈ ϝ , such that μ ≤ ⋁ j ∈ ϝ λ j , there is a finite subset ϝ ∘ of ϝ , such that μ ≤ ⋁ j ∈ ϝ ∘ λ j .
Lemma 4.1.
Let ( U , η , η * ) be a d f t s , r ∈ I ∘ , and s ∈ I 1 . If μ ∈ I U is ( r , s ) -fuzzy GS -compact, then μ is ( r , s ) -fuzzy compact.
Proof. 
Follows from Definitions 4.1 and 4.2. □
Theorem 4.1.
Let h : ( U , τ , τ * ) → ( V , η , η * ) be a DFGS -continuous mapping, r ∈ I ∘ , and s ∈ I 1 . If μ ∈ I U is ( r , s ) -fuzzy GS -compact, then h ( μ ) is ( r , s ) -fuzzy compact.
Proof. 
Let { λ j ∈ I V | η ( λ j ) ≥ r and η * ( λ j ) ≤ s } j ∈ ϝ with h ( μ ) ≤ ⋁ j ∈ ϝ λ j , then { h − 1 ( λ j ) ∈ I U | h − 1 ( λ j ) is ( r , s ) − g f s o } (by h is DFGS -continuous), such that μ ≤ ⋁ j ∈ ϝ h − 1 ( λ j ) . Since μ is ( r , s ) -fuzzy GS -compact, there is a finite subset ϝ ∘ of ϝ , such that μ ≤ ⋁ j ∈ ϝ ∘ h − 1 ( λ j ) . Thus, h ( μ ) ≤ ⋁ j ∈ ϝ ∘ λ j . Hence, the proof is completed. □
Definition 4.3.
Let ( U , η , η * ) be a d f t s , r ∈ I ∘ , and s ∈ I 1 , then μ ∈ I U is said to be an ( r , s ) -fuzzy almost compact iff for each family { λ j ∈ I U | η ( λ j ) ≥ r and η * ( λ j ) ≤ s } j ∈ ϝ , such that μ ≤ ⋁ j ∈ ϝ λ j , there is a finite subset ϝ ∘ of ϝ , such that μ ≤ ⋁ j ∈ ϝ ∘ C η , η * ( λ j , r , s ) .
Definition 4.4.
Let ( U , η , η * ) be a d f t s , r ∈ I ∘ , and s ∈ I 1 , then μ ∈ I U is said to be an ( r , s ) -fuzzy almost GS -compact iff for each family { λ j ∈ I U | λ j is ( r , s ) − g f s o } j ∈ ϝ , such that μ ≤ ⋁ j ∈ ϝ λ j , there is a finite subset ϝ ∘ of ϝ , such that μ ≤ ⋁ j ∈ ϝ ∘ C η , η * ( λ j , r , s ) .
Lemma 4.2.
Let ( U , η , η * ) be a d f t s , r ∈ I ∘ , and s ∈ I 1 . If μ ∈ I U is ( r , s ) -fuzzy almost GS -compact, then μ is ( r , s ) -fuzzy almost compact.
Proof. 
Follows from Definitions 4.3 and 4.4. □
Lemma 4.3.
Let ( U , η , η * ) be a d f t s , r ∈ I ∘ , and s ∈ I 1 . If μ ∈ I U is ( r , s ) -fuzzy compact (resp., GS -compact), then μ is ( r , s ) -fuzzy almost compact (resp., almost GS -compact).
Proof. 
Follows from Definitions 4.1, 4.2, 4.3 and 4.4. □
Remark 4.1.
The converse of Lemma 4.3 may not be true, as shown by Example 4.1.
Example 4.1.
Let V = I , k ∈ N − { 1 } , and ρ , λ k ∈ I V defined as follows:
ρ ( v ) = 1 , if v = 0 , 1 2 , otherwise ,
λ k ( v ) = 0.8 , if v = 0 , k v , if 0 < v ≤ 1 k , 1 , if 1 k < v ≤ 1 .
Also, ( η , η * ) defined on V as follows:
η ( μ ) = 1 , if μ ∈ { 0 ̲ , 1 ̲ } , 2 3 , if μ ≤ ρ , k k + 1 , if μ ≤ λ k , 0 , otherwise ,
η * ( μ ) = 0 , if μ ∈ { 0 ̲ , 1 ̲ } , 1 3 , if μ ≤ ρ , 1 k + 1 , if μ ≤ λ k , 1 , otherwise .
Thus, V is ( 1 2 , 1 2 ) -fuzzy almost compact, but it is not ( 1 2 , 1 2 ) -fuzzy compact.
Theorem 4.2.
Let h : ( U , τ , τ * ) → ( V , η , η * ) be a DF -continuous mapping, r ∈ I ∘ , and s ∈ I 1 . If μ ∈ I U is ( r , s ) -fuzzy almost GS -compact, then h ( μ ) is ( r , s ) -fuzzy almost compact.
Proof. 
Let { λ j ∈ I V | η ( λ j ) ≥ r and η * ( λ j ) ≤ s } j ∈ ϝ with h ( μ ) ≤ ⋁ j ∈ ϝ λ j , then { h − 1 ( λ j ) ∈ I U | h − 1 ( λ j ) is ( r , s ) − g f s o } (by h is DFGS -continuous), such that μ ≤ ⋁ j ∈ ϝ h − 1 ( λ j ) . Since μ is ( r , s ) -fuzzy almost GS -compact, there is a finite subset ϝ ∘ of ϝ , such that μ ≤ ⋁ j ∈ ϝ ∘ C τ , τ * ( h − 1 ( λ j ) , r , s ) . Since h is DF -continuous mapping, it follows
μ ≤ ⋁ j ∈ ϝ ∘ C τ , τ * ( h − 1 ( λ j ) , r , s )
≤ ⋁ j ∈ ϝ ∘ h − 1 ( C η , η * ( λ j , r , s ) )
= h − 1 ( ⋁ j ∈ ϝ ∘ C η , η * ( λ j , r , s ) ) .
Thus, h ( μ ) ≤ ⋁ j ∈ ϝ ∘ C η , η * ( λ j , r , s ) . Hence, the proof is completed. □
Definition 4.5.
Let ( U , η , η * ) be a d f t s , r ∈ I ∘ , and s ∈ I 1 , then μ ∈ I U is said to be an ( r , s ) -fuzzy nearly compact iff for each family { λ j ∈ I U | η ( λ j ) ≥ r and η * ( λ j ) ≤ s } j ∈ ϝ , such that μ ≤ ⋁ j ∈ ϝ λ j , there is a finite subset ϝ ∘ of ϝ , such that μ ≤ ⋁ j ∈ ϝ ∘ I η , η * ( C η , η * ( λ j , r , s ) , r , s ) .
Definition 4.6.
Let ( U , η , η * ) be a d f t s , r ∈ I ∘ , and s ∈ I 1 , then μ ∈ I U is said to be an ( r , s ) -fuzzy nearly GS -compact iff for each family { λ j ∈ I U | λ j is ( r , s ) − g f s o } j ∈ ϝ , such that μ ≤ ⋁ j ∈ ϝ λ j , there is a finite subset ϝ ∘ of ϝ , such that μ ≤ ⋁ j ∈ ϝ ∘ I η , η * ( C η , η * ( λ j , r , s ) , r , s ) .
Lemma 4.4.
Let ( U , η , η * ) be a d f t s , r ∈ I ∘ , and s ∈ I 1 . If μ ∈ I U is ( r , s ) -fuzzy nearly GS -compact, then μ is ( r , s ) -fuzzy nearly compact.
Proof. 
Follows from Definitions 4.5 and 4.6. □
Lemma 4.5.
Let ( U , η , η * ) be a d f t s , r ∈ I ∘ , and s ∈ I 1 . If μ ∈ I U is ( r , s ) -fuzzy compact (resp., GS -compact), then μ is ( r , s ) -fuzzy nearly compact (resp., nearly GS -compact).
Proof. 
Follows from Definitions 4.1, 4.2, 4.5 and 4.6. □
Remark 4.2.
The converse of Lemma 4.5 may not be true, as shown by Example 4.2.
Example 4.2.
Let V = I , 0 < k < 1 , and ν , ρ , λ k ∈ I V defined as follows:
ν ( v ) = 1 2 , if 0 ≤ v < 1 , 1 , if v = 1 ,
ρ ( v ) = 1 , if v = 0 , 1 2 , if 0 < v ≤ 1 ,
λ k ( v ) = v k , if 0 ≤ v ≤ k , 1 − v 1 − k , if k < v ≤ 1 .
Also, ( η , η * ) defined on V as follows:
η ( μ ) = 1 , if μ ∈ { ν , ρ , 0 ̲ , 1 ̲ } , max ( { 1 − k , k } ) , if μ = λ k , 0 , otherwise ,
η * ( μ ) = 0 , if μ ∈ { ν , ρ , 0 ̲ , 1 ̲ } , min ( { k , 1 − k } ) , if μ = λ k , 1 , otherwise .
Thus, V is ( 1 2 , 1 2 ) -fuzzy nearly compact, but it is not ( 1 2 , 1 2 ) -fuzzy compact.
Theorem 4.3.
Let h : ( U , τ , τ * ) → ( V , η , η * ) be a DF -continuous and DF -open mapping, r ∈ I ∘ , and s ∈ I 1 . If μ ∈ I U is ( r , s ) -fuzzy nearly GS -compact, h ( μ ) is ( r , s ) -fuzzy nearly compact.
Proof. 
Let { λ j ∈ I V | η ( λ j ) ≥ r and η * ( λ j ) ≤ s } j ∈ ϝ with h ( μ ) ≤ ⋁ j ∈ ϝ λ j , then { h − 1 ( λ j ) ∈ I U | h − 1 ( λ j ) is ( r , s ) − g f s o } (by h is DFGS -continuous), such that μ ≤ ⋁ j ∈ ϝ h − 1 ( λ j ) . Since μ is ( r , s ) -fuzzy nearly GS -compact, there is a finite subset ϝ ∘ of ϝ , such that μ ≤ ⋁ j ∈ ϝ ∘ I τ , τ * ( C τ , τ * ( h − 1 ( λ j ) , r , s ) , r , s ) . Since h is DF -continuous and DF -open, it follows
h ( μ ) ≤ ⋁ j ∈ ϝ ∘ h ( I τ , τ * ( C τ , τ * ( h − 1 ( λ j ) , r , s ) , r , s ) )
≤ ⋁ j ∈ ϝ ∘ I η , η * ( h ( C τ , τ * ( h − 1 ( λ j ) , r , s ) ) , r , s )
≤ ⋁ j ∈ ϝ ∘ I η , η * ( h ( h − 1 ( C η , η * ( λ j , r , s ) ) ) , r , s )
≤ ⋁ j ∈ ϝ ∘ I η , η * ( C η , η * ( λ j , r , s ) , r , s ) .
Hence, the proof is completed. □
Lemma 4.6.
Let ( U , η , η * ) be a d f t s , r ∈ I ∘ , and s ∈ I 1 . If μ ∈ I U is ( r , s ) -fuzzy soft nearly GS -compact (resp., nearly compact), then μ is ( r , s ) -fuzzy soft almost GS -compact (resp., almost compact).
Proof. 
Follows from Definitions 4.3, 4.4, 4.5 and 4.6. □
Remark 4.3.
From the previous results and definitions, we can summarize the relationships among different types of fuzzy compactness as in the next diagram.
GS - compactness → compactness
↓ ↓
nearly GS - compactness → nearly compactness
↓ ↓
almost GS - compactness → almost compactness

5. Conclusion and Future Work

In this article, “ ( r , s ) - GFS -regular” and “ ( r , s ) - GFS -normal” spaces have been defined as two new notions of higher fuzzy separation axioms in double fuzzy topological space based on the sense of Šostak. Several properties and characterizations of these separation axioms have been obtained. Thereafter, we have introduced a novel class of generalizations of fuzzy closed subsets called “ ( r , s ) − g ⊛ f s c sets” and some characterizations have been discussed. In addition, we have defined new types of fuzzy mappings and the relationship between these mappings have been introduced with the help of some problems. Moreover, we have shown that
( r , s ) − f s c ⇒ ( r , s ) − g ⊛ f s c
⇓ ⇓
( r , s ) − s g f c ( r , s ) − g ⊝ f s c
⇓ ⇓
( r , s ) − g f s c
but in general, the converses of the above implications may not be true. In the end, several new types of fuzzy compactness in the frame of double fuzzy topologies have been introduced and some properties have been given. Also, the relationship between them have been explored.
In the upcoming papers, we shall discuss the concepts given here in the frames of a fuzzy idealization [42,43] and fuzzy soft r-minimal structures [44,45]. Moreover, we will study the main properties of classical compactness in the frame of double fuzzy topologies such as the relationship between closed and compact subsets.

Data Availability Statement

No data were used to support this study.

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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