In the present paper, we have continued to study the properties of semi-separation axioms in ditopological texture spaces. We prove that the semi-Ti-spaces () are productive in ditopological spaces. Finally, semi-separation axioms are given for soft fuzzy topological spaces by using the categorical isomorphism between soft fuzzy topologies and ditopologies.
The concept of soft set theory was introduced by Molodtsov [21] as a new approach for modeling uncertainties. The relationship between soft sets and information systems was discussed in [22]. Later, the idea was expanded soft fuzzy set by Maji et al. [20]. Shabir and Naz [23] gave the concept of soft topological spaces and studied soft neighborhood of a point, soft separation axioms. The concept of soft fuzzy set over a poset was introduced by Tiryaki [24]. On the other hand, initial soft topologies were given in [1]. In recent studies, the notions of semi-open sets with related properties [9] were given in soft topological spaces.
The theory of texture spaces is an alternative setting for fuzzy sets and therefore, many properties of Hutton algebras (known as fuzzy lattices) can be discussed in terms of textures [2-7]. Ditopologies (dichotomous topologies) on textures unify the fuzzy topologies, topologies and bitopologies in a non-complemented setting by means of duality in the textural concepts [6]. A texturing is a family of subsets of a given universe S satisfying certain conditions which are related to the properties of the power set . Then the pair is called a texture space.
It is studied Ti-spaces (), regularity and normality- separation axioms in ditopological texture spaces in [7]. On the other hand, semi-regularity axioms are introduced and studied in [17].
Now let X be a set and μ an -fuzzy subset of X, M ⊆ X. Then the pair (μ, M) is said to be a soft fuzzy subset of X. The set of all soft fuzzy subsets of X is denoted by SF (X). It was proved in [24] that SF (X) is a Hutton algebra and corresponding Hutton texture is isomorphic to the textural product of the discrete texture and the unit interval texture where . It is defined some which are used morphisms mappings between soft fuzzy lattices associated with difunctions between textures. Furthermore, it was shown that the category SF-Top of soft fuzzy topologies and continuous mappings is isomorphic to the category SF-Ditop of ditopologies on the product texture and bicontinuous difunctions. As result of this isomorphism, the notions of separation axioms were introduced in the soft fuzzy topological spaces [24].
The first aim of this study is to present a discussion on the concepts of semi-separation axioms which may give more suitable environments for some areas. Secondly, using the above mentioned isomorphisms, we give a new approach for these concepts in soft fuzzy topological spaces with textural view.
This paper is organized into eight sections: the next section contains a review of well-known properties of ditopological texture spaces, and the concept of semi-open set and semi-regularity axioms are recalled. The Section 3 is devoted to the soft topological spaces, and after this section it is given the lattice of soft fuzzy sets. In the Sections 5 and 6, it is introduced the point semi-separation axioms and further semi-regularity axioms, respectively. Finally, soft fuzzy topological spaces and semi-separation axioms are given by using the categorical isomorphism between soft fuzzy topologies and ditopologies in the Sections 7 and 8.
Preliminaries
In this section, we present the basic definitions and results of the theory of ditopological texture spaces which is needed in the sequel [2–7]. Moreover, in [14] and [17], semi-open and semi-closed sets were introduced in view of ditopological texture spaces.
Texture Space: If S is a set, a texturing is a point-separating, complete, completely distributive lattice containing S and ∅ and for which meet coincides with intersection and finite joins with union. If is textured by S, then we call a texture space or shortly, texture.
For a texture , most properties are appropriately defined in terms of the p-sets and, as a dually, the q-sets, .
Complementation: In general, a texturing of S need not be closed under set complementation, however there exists a mapping is said to be a complementation if σ (σ (A)) = A, and A ⊆ B implies σ (B) ⊆ σ (A) for . A texture with a complementation is called a complemented texture.
Example 2.1. Some examples of texture spaces,
The pair is called the discrete texture on X where is the power set of X. Obviously, for all x ∈ X we have Px = {x}, Qx = X \ {x} and is the ordinary complementation on defined by π (Y) = X \ Y for all Y ⊆ X. Therefore, is plain.
Let L = (0, 1], and λ ((0, r]) = (0, 1 - r], r ∈ [0, 1]. Then is complemented texture space. Here Pr = Qr = (0, r] for all r ∈ L. Hence, is not plain.
For define , ι ([0, t]) = [0, 1 - t) and ι ([0, t)) = [0, 1 - t], t ∈ [0, 1]. is a complemented texture, which we refer to as the unit interval texture. Here Pt = [0, t] and Qt = [0, t) for all . The natural ditopology is given by and .
If , are textures, the product texturing of S × T consists of arbitrary intersections of sets of the form (A × T) ∪ (S × B), , and is called the product of and . For s ∈ S, t ∈ T we obviously have P(s,t) = Ps × Pt and Q(s,t) = (Qs × T) ∪ (S × Qt).
Ditopological Texture Space: A ditopology on a texture is a pair (τ, κ) of subsets of , where the set of open setsτ and the set of closed setsκ verifies
Therefore a ditopology is essentially a “topology” for which there is no a priori relation between the open and closed sets.
For we define the closurecl (A) and the interiorint (A) of A under (τ, κ) by the equalities
If (τ, κ) is a ditopology on a complemented texture we say (τ, κ) is complemented if κ = σ (τ). In this case we have σ (cl (A)) = int (σ (A)) and σ (int (A)) = cl (σ (A)).
Product Texture Space: If , j ∈ J, are textures, and for some k ∈ J we write
Then the product texturing on S consists of arbitrary intersections of elements of the set
Product Ditopological Texture Spaces: Let be ditopological texture spaces and the product of the textures . For each j ∈ J,
define the j-th projection difunction (πj, Πj) on is called the product ditopology on [6]. Note that the product ditopology is the coarsest ditopology making the projection difunctions (πj, Πj), j ∈ J are bicontinuous.
Sum Ditopological Texture Spaces: Let i ∈ I be textures with Si∩ Sj = ∅ for i ≠ j. Let S = ⋃ i∈ISi and . Then is a texture which is called sum of disjoint textures i ∈ I and if are the complemented textures for all i ∈ I then the complementation
makes is a complemented texture.
We have noted that by [6]; If is a complemented sum texture of , then for j ∈ I it can be obtained the following equalities:
Ps = Psj × { j } ,
Qs = (Qsj × { j }) ∪ (⋃ k∈I∖{j}Sk × { k }) .
Direlation: Let us consider the product texture of the texture spaces and and denote the p-sets and the q-sets by and , respectively. Obviously, and where t ∈ T and s ∈ S. Hence
is said to be a relation fromto if it satisfies
.
satisfying Ps ⊈ Qs′ and .
is said to be a corelation from to if it satisfies
.
satisfying Ps′ ⊈ Qs and .
A pair (r, R), where r is a relation and R a corelation from to , is said to be a direlation from to .
One of the most useful notions of ditopological texture spaces is that of difunction. A difunction is a special type of direlation.
Difunction: Let (f, F) be a direlation from to . Then (f, F) is said to be a difunction fromto if it satisfies the following two conditions.
For s, s′ ∈ S, Ps ⊈ Qs′⇒ ∃ t ∈ T with and .
For t, t′ ∈ T and s ∈ S, and .
Definition 2.2. Let be a difunction. For and , the A-sections and the B-presections with respect to (f, F) are given as follows:
respectively.
For a given difunction, the inverse image and the inverse co-image are equal; and the image and co-image are usually not.
We note that ((f←) ←) (A) = f→ (A) and ((F←) ←) (A) = F→ (A) by [5, Lemma 2.9].
Definition 2.3. Let be a difunction. Then (f, F) is said to be surjective if it satisfies the condition
SUR. For t, t′ ∈ T, Pt ⊈ Qt′⇒ ∃ s ∈ S with and .
Likewise, (f, F) is said to be injective if it satisfies the condition
INJ. For s, s′ ∈ S and t ∈ T, and .
If (f, F) is both injective and surjective then it is said to be bijective.
Remark 2.4. As usual, difunctions are not directly related to ordinary (point) mappings between the base sets, but we recall from [5, Lemma 3.4] that if are textures and φ : S → T an ω-compatible point mapping, namely one satisfying that if Ps ⊈ Qs′ then Pφ(s) ⊈ Qφ(s′), hence the equalities
define a difunction (fφ, Fφ) on to . Moreover, for we have , where φ←B = ⋁ {Ps ∣ Pφ(u) ⊆ B, ∀u ∈ S with Ps ⊈ Qu}.
We will denote by ifTex the construct of textures and ω-preserving mappings between them.
Semi-open sets and semi-closed sets
Definition 2.5. Let be ditopological space. Then is called
semi-open if there is G ∈ τ such that G ⊆ A ⊆ cl (G) or equivalently if A ⊆ cl (int (A)).
semi-closed if there is K ∈ κ such that int (K) ⊆ A ⊆ K or equivalently if int (cl (A)) ⊆ A.
The class of all semi-open (respectively semi-closed) sets is denoted by SO (S) (respectively SC (S)).
Definition 2.6. [14] Let be a ditopological space and A∈ . We define
The semi-interior sint (A) of A is the set
The semi-closure scl (A) of A is the set
Obviously, for a complemented ditopological texture space : is semi-open iff σ (A) is semi-closed.
Definition 2.7. [14] Let j = 1, 2 be ditopological spaces. A difunction is said to be
semi-continuous (semi-irresolute) if for each open (semi-open) set , the inverse image F← (A) is a semi-open set.
semi-cocontinuous (semi-co-irresolute) if for each closed (semi-closed) set , the inverse image f← (B) is a semi-closed set.
semi-bicontinuous (semi-bi-irresolute) if it is semi-continuous and semi-cocontinuous (semi-irresolute and semi-co-irresolute).
Before leaving this section, let us recall semi-R0, semi-R1 and semi-regular axioms in ditopological texture spaces in [17].
Semi-regularity axioms
Definition 2.8. Let be a ditopological texture space. Then (τ, κ) is called
semi-R0 if A ∈ SO (S) , A ⊈ Qs⇒scl (Ps) ⊆ A, and
semi-co-R0 if B ∈ SC (S) , Ps ⊈ B⇒B ⊆ sint (Qs).
Definition 2.9. Let be a ditopological texture space. Then (τ, κ) is called
semi-R1 if A ∈ SO (S) , A ⊈ Qs, Pt ⊈ A⇒ ∃ G ∈ SO (S) , G ⊈ Qs, Pt ⊈ scl (G), and
semi-co-R1 if B ∈ SC (S) , Ps ⊈ B, B ⊈ Qt⇒ ∃ K ∈ SC (S) , Ps ⊈ K, sint (K) ⊈ Qt.
Definition 2.10. Let be a ditopological texture space. Then (τ, κ) is called
semi-regular if A ∈ SO (S) , A ⊈ Qs⇒ ∃ G ∈ SO (S) , G ⊈ Qs, scl (G) ⊆ A, and
semi-co-regular if B ∈ SC (S) , Ps ⊈ B⇒ ∃ K ∈ SC (S) , Ps ⊈ K, B ⊆ sint (K).
An overview of soft topological spaces
Definition 3.1. [21] Let X be an universe and E be a set of parameters. Let denote the power set of X and A be a non-empty subset of E. Then a pair (F, A) is called a soft set over X, where F is a set-valued mapping given by .
By the definition, a soft set over X is a parameterized family of subsets of X. For a particular e ∈ A, F (e) is the set of e-approximate elements of the soft set (F, A) . Moreover, a soft set over X may be represented by the set of ordered pairs
A soft set (F, A) over X is said to be
a null soft set and denoted by if F (e) = ∅ , for each e ∈ A .
an absolute soft set and denoted by , if F (e) = X, for each e ∈ A .
Definition 3.2. [23] A soft topology τ is a family of soft sets over X satisfying the following axioms:
∈τ .
The union of any family of soft sets in τ belongs to τ .
The intersection of any two soft sets in τ belongs to τ .
The triplet (X, τ, E) is said to be a soft topological space. Every member of τ is called soft open set.
It is known that a fuzzy set μ on X is characterized by a mapping with domain X and values in [0, 1) i.e. μ : X → [0, 1). The value μ (x) represents the membership degree of x ∈ X in the fuzzy set μ, for x ∈ X . Let denotes the family of all fuzzy sets on X where Then
Definition 3.3. [20] Let A ⊆ E. A pair (μ, A) is called fuzzy soft set over X, where μ is a mapping from the parameter set E into i.e., such that μ (e) ≠ 0X, if e ∈ A and μ (e) = 0X, otherwise, where 0X denotes the empty fuzzy set on X .
Then a fuzzy soft set (μ, A) is defined as follows:
where the symbol “a ={ xμa(x) ∣ x ∈ X }” indicates that the membership degree of the element x ∈ X is μa (x) where μa : A→ is the membership function of the fuzzy set μ (a) .
denotes the family of fuzzy soft sets over X . Then
is said to be a null fuzzy soft set and denoted by if μ (e) = ∅ X for each e ∈ E .
is said to be an absolute fuzzy soft set and denoted by , if μ (e) =1X for each e ∈ E .
Definition 3.4. A fuzzy soft topological space on a non-empty set X is a pair (X, τ) where τ denotes a family of fuzzy soft sets over X satisfying the following axioms:
∈τ .
The union of any family of fuzzy soft sets in τ belongs to τ .
The intersection of any two fuzzy soft sets in τ belongs to τ .
The triplet (X, τ, E) is said to be a fuzzy soft topological space over X and every member of τ is called fuzzy soft open set in X .
It was proved in [3] that F (X) which is fuzzy subsets of X is isomorphic to texturing where Now let [0, r) ∣ r ∈ [0, 1]} be the unit interval texture. As given detailed in the next section, it is shown that [24] the element of the product texturing can be presented as pairs (μ, M) , where μ ∈ F (X) and For x ∈ X, it is described that if x ∈ M then μ (x) is realized or hard value, otherwise it is soft or unrealized. So the pairs (μ, M) are said to be soft fuzzy sets.
Soft fuzzy sets are shown to have a richer mathematical theory than classical fuzzy sets. Moreover, soft fuzzy topologies on X are specialized the notion of topologies on X . Then we have a considerable simplification arising from the very clean point structure.
Lattice of soft fuzzy sets
We begin by recalling [25] that if (N, ≤) is a poset (i.e. partially ordered set) then the set of lower set of N is a plain texturing of N. If (X, ≤) is also poset and we take X × N such that the product order x1 ≤ x2 and n1 ≤ n2 ⇔ (x1, n1) ≤ (x2, n2), then denotes the corresponding texturing of X × N, and according to [25, Proposition 4.1]t1 the texture is the product of the texture spaces and . The elements of may be regarded as -fuzzy subsets of X. Indeed, if we set
then by [25, Proposition 4.2]t1 the mapping
is an isomorphism from FN (X) to the texturing of U × N such that inverse A → μ (A), μA (x) ={ n ∣ (x, n) ∈ A }.
On the other hand, if n → n′ is an order-reversing involution on N then a complementation σ satisfies σ (Pn) = Qn′, n ∈ N, and conversely [25, Theorem 2.10]t1. Note that μ ∈ FN (X) define μ′ (x) = { n ∈ N ∣ n′ ∉ μ (x′) } . Hence the mapping μ → μ′ is an order-reversing involution on FN (X). Also (x, n) → (x, n) ′ = (x′, n′) is an order-reversing involution on (X × N, ≤), and the corresponding complementation σX×N on is the product σX ⊗ σN of σX and σN . For μ∈ FN (X), we have Aμ ={ (x, n) ∣ x ∈ U, n ∈ N, n′ ∈ μ (x′) } and σU×N (Aμ) = Aμ′ .
As a result, the mapping μ → Aμ becomes an isomorphism between the Hutton algebras FN (X) and .
Now we take the set X with the discrete ordering x1 = x2 ⇔ x1 ≤ x2 and the trivial involution x → x, that is x′ = x for all x ∈ X. Then the corresponding texture is the complemented discrete texture which is given Examples 2.1 (i). Secondly, we consider with the usual ordering and the order reversing involution t → 1 - t, . Then, the corresponding texture is unit interval texture which is given in Examples 2.1 (iii). This gives us the family of -fuzzy subsets of X, and the textural representation . Here, since X has discrete ordering we just have . Then for x ∈ X we have η (x) = Pt or η (x) = Qt for some .
Note that the textural presentation of the Hutton algebra F (X) of classical fuzzy sets (that is -fuzzy subsets of X) is the product texture where is given in Examples 2.1 (ii).
In [24], the element of is represent as pairs (μ, M), where and μ ∈ F (X). For the benefit of the reader, we give on the motivation and we recall all the necessary basic concepts and results considering the soft fuzzy lattice and mappings between soft fuzzy lattices.
Definition 4.1. Let X be a nonempty set. Let μ be an -fuzzy subset of X such that μ : X → [0, 1) and M ⊆ X. Hence, the ordered pair (μ, M) is said to be as a soft fuzzy set in X. The family of all soft fuzzy subsets of X is denoted as SF (X) .
A mapping from to SF (X) is given by η → (η1, η2) where
Conversely if (μ, M) ∈ SF (X) then we may set ξ (μ, M) = η where
Lemma 4.2.The mapping ξ : SF (X) → defined above is a bijection with inverse ξ-1 given by ξ-1 (η) = (η1, η2).
A simple calculation shows that
Definition 4.3. The relation ⊑ on SF (X) is given by (μ, M) ⊑ (v, N) ⇔ Aξ(μ,M) ⊆ Aξ(v,N).
We have an order-preserving bijection between (SF (X) , ⊑) and because inclusion is a partial order on .
Proposition 4.4.Let
has a meet in (SF (X) , ⊑), denoted by ⊓j∈J(μj, Mj) and considered as
where μ (x) = ⋀ j∈Jμj (x) and
for all x ∈ X.
has a join in (SF (X) , ⊑), denoted by ⊔j∈J(μj, Mj) and considered as
where μ (x) = ⋁ j∈Jμj (x) and
for all x ∈ X.
Corollary 4.5.For all (μj, Mj) ∈ SF (X) , j ∈ J, we have
(SF (X) , ⊑) is a complete lattice since is a complete lattice. In particular, ξ is an isomorphism between (SF (X) , ⊑) and .
Definition 4.6. Let X be a nonempty set. Then,the complement of (μ, M) ∈SF (X) is defined as (μ, M) ′ = (1 - μ, X ∖ M) .
In particular, (SF (X) , ⊑) is a Hutton Algebra since the complement ′ on SF (X) is idempotent and order-reversing. Furthermore, (SF (X) , ⊑) is isomorphic to , and hence to .
In this study, the product texture space will denoted by , that is
By the definition of πX ⊗ ι,
whenever (μ, M) ∈ SF (X) .
Now consider the soft fuzzy points (xs, { x }) and the soft fuzzy copoint (xs, X ∖ { x }) in SF (X) defined by
where base x ∈ X and value We denote (xs, { x }) ⊑ (μ, M) by (xs, { x }) ∈ (μ, M) , and refer to (xs, { x }) as an element of (μ, M) .
The mappings between soft fuzzy sets
Now let us consider sets X, Y with the discrete order and a point function φ : X → Y. Thus <φ, id > : ZX → ZY defined by <φ, id > (x, s) = (φ (x) , s) is ω-preserving which is given in Remark 2.4, regarded as a mapping from to where id is the identity on . Let us denote by SF-Set the construct whose objects are pairs (X, SF (X) ) and morphisms from Set. Specially we define SF-Set→ifTex by setting
Obviously, SF-Set is embedded as an isomorphism-closed sub construct of ifTex.
The point function φ is used to define mapping between SF (X) to SF (Y) . To conclude this section we look at the difunction (f, F) corresponding to <φ, id > (x, s) = (φ (x) , s) as in Remark 2.4. Since textures are plain, we have
The image and co-image operators map from to , and inverse image and inverse co-image operators, which are equal, map from to . In view of the isomorphism between SF (X) and , and that between SF (Y) and , these lead to the required mappings, as detailed follows:
Proposition 4.7.Let φ : X → Y be a point function. The mapping φ← from SF (Y) to SF (X) corresponding to the inverse image and inverse co-image of the difunction (f, F) is given by
Point semi-separation propertiesin ditopological spaces
Before presenting our definition of the semi-T0 property we give two general theorems which will be found useful.
Theorem 5.1.Let be ditopological space. If contain S, ∅ and be closed under arbitrary joins. Then the following are equivalent:
A = ⋂ j∈JAj for all and,j ∈ J .
For all s, t ∈ S, if Ps ⊈ Pt then there exists satisfying Pt ⊆ A ⊆ Qs .
For all s, t ∈ S, if Qs ⊈ Qt then there exists satisfying Pt ⊆ A ⊆ Qs .
Proof. (i) ⇒ (ii) Take Pt = A . For some ,j ∈ J we have Ps ⊈ Pt = ⋂ j∈JAj . Therefore, for some j ∈ J, Aj ⊆ Qs and Pt ⊆ Aj .
(ii) ⇒ (iii) Consider Qs ⊈ Qt. According to definition, Qs =⋁ { Pu ∣ Ps ⊈ Pu } so we have u ∈ S with Ps ⊈ Pu and Pu ⊆ Qs. We can write Pu = A, thus we have Pt ⊆ A, whence Pt ⊆ A ⊆ Qs.
(iii) ⇒ (i) Consider , hence we obtain A =⋂ { Qs ∣ s ∉ A }, for all . □
Theorem 5.2.Let be ditopological space. If contain S, ∅ and be closed under arbitrary intersections. Then the following are equivalent:
B = ⋁ j∈JBj for all and, j ∈ J .
For s, t ∈ S, if Qs ⊈ Qt then there exists satisfying Ps ⊈ B ⊈ Qt .
For s, t ∈ S, if Ps ⊈ Pt then there exists satisfying Ps ⊈ B ⊈ Qt .
Proof. Dual to the proof of Theorem 5.1, and we are omitted.□
Let (SC (S) ∪ SO (S)) ∨ denote the set of arbitraryjoins of semi-open and semi-closed sets in (SC (S) ∪ SO (S)), and (SC (S) ∪ SO (S)) ∩ denote the set of arbitrary intersections of semi-open and semi-closed sets in (SC (S) ∪ SO (S)) .
Definition 5.3. A ditopological space which satisfies the equivalent conditions obtained by setting
in Theorem 5.1, and
in Theorem 5.2
is called a semi-T0-space.
Theorem 5.4.The following are characteristic properties of semi-T0-space ditopological texture spaces:
Ps ⊈ Pt ⇒ ∃ Cj ∈ (SC (S) ∪ SO (S)), j ∈ J satisfying Pt ⊆ ⋁ j∈JCj ⊆ Qs .
Qs ⊈ Qt ⇒ ∃ Cj ∈ (SC (S) ∪ SO (S)), j ∈ J satisfying Pt ⊆ ⋂ j∈JCj ⊆ Qs .
Qs ⊈ Qt ⇒ ∃ C ∈ (SC (S) ∪ SO (S)) satisfying Ps ⊈ C ⊈ Qt .
scl (Ps) ⊆ scl (Pt) and sint (Qs) ⊆ sint (Qt) ⇒ Qs ⊆ Qt .
Proof. We need only verify (iv) since the other conditions follow from Theorem 5.1 with
, or from Theorem 5.2 with . If we assume that Qs ⊈ Qt, then C ∈ (SC (S) ∪ SO (S)) satisfies Ps ⊈ C ⊈ Qt by (iii). If C ∈ SC (S) then C ⊈ Qt ⇒ Pt ⊆ C ⇒ scl (Pt) ⊆ C, and we have scl (Ps) ⊈ scl (Pt) . If C ∈ SO (S) then Ps ⊈ C ⇒ C ⊆ Qs ⇒ C ⊆ sint (Qs), and so sint (Qs) ⊈ sint (Qt) . Hence (iii)⇒(iv).
Conversely, if (iv) satisfied and Ps ⊈ Pt then scl (Ps) ⊈ scl (Pt) or sint (Qs) ⊈ sint (Qt) . In the first case, Pt ⊆ scl (Pt) ⊆ Qs and in the other case Pt ⊆ sint (Qs) ⊆ Qs. Therefore (iv)⇒(i).□
Now, we give the semi-T1 axiom.
Definition 5.5. A ditopological space is said to be:
semi-T1-space if it is semi-T0 and semi-R0 .
semi-co-T1-space if it is semi-T0 and semi-co–R0 .
Theorem 5.6.Let be a ditopological space.
is semi-T1-space iff it satisfies the conditions of Theorem 5.2 with In particular, the following are characteristic of a semi-T1 ditopological space.
For any we have Fi ∈ SC (S) , i ∈ I satisfies A = ⋁ i∈IFi .
For all s, t ∈ S, if Qs ⊈ Qt then there exists F ∈ SC (S) satisfying Ps ⊈ F ⊈ Qt .
is semi-co-T1-space iff it satisfies the conditions of Theorem 5.1 with In particular, the following are characteristic of a semi-co-T1 ditopological space.
For any we have Gi ∈ SO (S) , i ∈ I satisfies A = ⋂ i∈IGi .
For all s, t ∈ S, if Qs ⊈ Qt then there exists G ∈ SO (S) satisfying Ps ⊈ G ⊈ Qt .
Proof. We prove (1), leaving the dual proof of (2) to the interested reader. Let be semi-T0 and semi–R0 . Take s, t ∈ S with Qs ⊈ Qt . According to Theorem 5.4 (iii), we get C ∈ (SC (S) ∪ SO (S)) satisfying Ps ⊈ C ⊈ Qt . If C ∈ SC (S) then F = C ∈ SC (S) satisfies Ps ⊈ C ⊈ Qt . Otherwise, if C ∈ SO (S) then C ⊈ Qt implies scl (Pt) ⊆ C by semi-R0, whence for F = scl (Pt) we have Ps ⊈ C ⊈ Qt . This establishes Theorem 5.2 (ii) for Conditions (a) and (b) are obtained from (i) and (ii) of Theorem 5.2, respectively.
Conversely, let satisfy Theorem 5.2 with Then, it satisfies this theorem with , so is semi-T0. To show it is semi-T1 it remains to show it is semi-R0 . But by Theorem 5.2 (i) we may choose A = G ∈ SO (S) satisfying G = ⋁ i∈IFi, Fi ∈ SC (S) , hence the result follows by [17, Theorem 4.2 (a) (2)]. □
Corollary 5.7.For a complemented ditopological space the notions of semi-T1 and semi-co-T1coincide.
Proof. This is obvious by [17, Corollary 4.3]. Indeed, if A = ⋁ j∈JFj, Fj ∈ SC (S), then σ (A) = ⋂ j∈Jσ (Fj) , σ (Fj) ∈ SO (S), and thus the proof is clear.□ Let us now consider the semi-T2 axiom. Thus, we have the following.
Definition 5.8. A ditopological space is said to be:
semi-T2-space if it is semi-T0 and semi-R1 .
semi-co-T2-space if it is semi-T0 and semi-co-R1 .
Theorem 5.9.The following are equivalent for a ditopology
is semi-T2 and semi-co-T2 .
For all s, t ∈ S, if Qs ⊈ Qt then there exists H ∈ SO (S) and K ∈ SC (S) satisfying H ⊆ K, Ps ⊈ K and H ⊈ Qt .
For there exists j ∈ J, i ∈ Ij satisfying for all i, j and
Proof. (1) ⇒ (2) Assume that Qs ⊈ Qt. Since is semi-T0 we have C ∈ (SC (S) ∪ SO (S)) with Ps ⊈ C ⊈ Qt by Theorem 5.4 (iii). If C ∈ SO (S), then H ∈ SO (S) satisfies Ps ⊈ scl (H) and H ⊈ Qt by the semi-R1 axiom. Thus (2) is verified for K = scl (H). Otherwise, if C ∈ SC (S) then K ∈ SC (S) satisfies Ps ⊈ K, sint (K) ⊈ Qt by the semi-co-R1 axiom. Thus (2) is verified for sint (K) = H .
(2) ⇒ (3) We get A =⋂ { Qs ∣ Ps ⊈ A } = ⋁ { Pt ∣ A ⊈ Qt } for For s, t verifying A ⊈ Qt and Ps ⊈ A we have Qs ⊈ Qt and so there exists with and , We obviously have
(3) ⇒ (1) Left to the reader.□
Proposition 5.10.For a complemented ditopological space the notions of semi-T2 and semi-co-T2 properties are equivalent.
Proof. Straightforward, because semi-R1 and semi-co-R1 properties are equivalent under complemented ditopological space.□
Remark 5.11. By definitions the following implications hold
semi-T2⇒semi-T1⇒semi-T0 and
semi-co-T2⇒semi-co-T1⇒semi-co-T0 .
Corollary 5.12.A semi-coregular T1 space is semi-T2 and semi-co-T2 and a regular semi-co-T1 space is semi-T2 and semi-co-T2 .
Proof. We prove the first result, leaving the dual proof of the other result to the interested reader. Let us assume that is semi-coregular and semi-T1 . Consider s, t ∈ S such that Qs ⊈ Qt. According to Theorem 5.6 (1)(b) we get F ∈ SC (S) such that Ps ⊈ F ⊈ Qt, thus we have K ∈ SC (S) such that Ps ⊈ K and F ⊆ sint (K) according to the definition of semi-coregularity. Now if we take sint (K) = H ∈ SO (S), then we obtain that H ⊈ Qt and H ⊆ K, thus is semi-T2 and semi-co-T2 by Theorem 5.9.□
Theorem 5.13.Let be the disjoint sum of non-empty disjoint ditopological texture spaces (), j ∈ J. Then is semi-T0 iff () is semi-T0 whenever j ∈ J.
Proof. Let us assume that each (), j ∈ J is semi-T0. Consider s, t ∈ S such that Qs ⊈ Qt . Let j, k ∈ J be the unique indices verifying s ∈ Sj, t ∈ Sk . If j ≠ k then , thus C = Sk verifies C ∈ (SC (S) ∪ SO (S)) and Ps ⊈ C ⊈ Qt . Otherwise, if j = k then and we have Ck ∈ (SC (S) ∪ SO (S)) k with In this case, C = Ck satisfies C ∈ (SC (S) ∪ SO (S)) and Ps ⊈ C ⊈ Qt . So is semi-T0 by Theorem 5.4 (iii).
The converse is omitted.□
Corollary 5.14.Let be the disjoint sum of non-empty disjoint ditopological texture spaces (), j ∈ J. Then
is semi-T1 (semi-co-T1) iff (Sj, τj, κj) is semi-T1 (semi-co-T1) whenever j ∈ J.
is semi-T2 (semi-co-T2) iff (Sj, τj, κj) is semi-T2 (semi-co-T2) whenever j ∈ J.
Proof. It is obvious from Theorems 5.13 and [17,Theorem 4.12].□
Theorem 5.15.Let be the product of non-empty disjoint ditopological texture spaces (, κj), j ∈ J . Then is semi-T0 iff (, κj) is semi-T0 whenever j ∈ J.
Proof. Let be semi-T0. Consider any j ∈ J and for sj, tj ∈ Sj with Qsj ⊈ Qtj . Choose so that k ∈ J ∖ { j } , since Sk ≠ ∅ . Now let t = (ti) and s = (si) ∈S defined by
It is imply that Qs ⊈ Qt, since Qsj ⊈ Qtj . By the Theorem 5.4 (iii), there exists C ∈ (SC (S) ∪ SO (S)) such that Ps ⊈ C ⊈ Qt . Firstly, assume the case C ∈ SO (S) . According to the definition of the product topology, we get j1, j2, …, jn ∈ J and Cjk ∈ SO (Sjk) , 1 ≤ k ≤ n, so that and . Thus we get k, 1 ≤ k ≤ n, for which Ps ⊈ E (jk, Cjk) ⊈ Qt, so Psjk ⊈ Cjk ⊈ Qtjk. This leads to an contradiction if jk ≠ j since then sjk = tjk . Then jk = j, and we obtain Cj ∈ SO (Sj) satisfying Psj ⊈ Cj ⊈ Qtj.
The other case, for C ∈ SC (S), can be handled similarly above. Thus each (), j ∈ J is semi-T0 by Theorem 5.4 (iii).
Conversely, let us assume that each (), j ∈ J is semi-T0. Consider t = (tj) and s = (sj) ∈ S such that Qs ⊈ Qt. According to Theorem 5.4 (iii), it will be sufficient to find C ∈ (SC (S) ∪ SO (S)) such that Ps ⊈ C ⊈ Qt. On the other hand because Qs = ⋃ j∈JE (j, Qsj) ⊈ Qt there exists j ∈ J such that E (j, Qsj) ⊈ Qt and so E (j, Qsj) ⊈ E (j, Qtj) , implying that Qsj ⊈ Qtj and by Theorem 5.4 (iii) we have Cj ∈ (SC (S) ∪ SO (S)) j such that Psj ⊈ Cj ⊈ Qtj . Obviously C = E (j, Cj) has the required properties. Thus is semi-T0 . □
Corollary 5.16.Let be the product of non-empty disjoint ditopological texture spaces (, κj), j ∈ J . Then
is semi-T1 (semi-co-T1) iff (, κj) is semi-T1 (semi-co-T1) whenever j ∈ J.
is semi-T2 (semi-co-T2) iff (, κj) is semi-T2 (semi-co-T2) whenever j ∈ J.
Proof. Clear by Theorems 5.15 and [17, Theorem 4.13].□
Further semi-separation axioms
Definition 6.1. A ditopological texture space is said to be semi-T3 (semi-co-T3) if it is semi-T0 and semi-regular (respectively, semi-co-regular).
Remark 6.2. By definitions the following implications hold
According to [17, Corollary 9]kd, we have
Proposition 6.3.For a complemented ditopological space the notions of semi-T3 and co-semi-T3 properties are equivalent.
Definition 6.4. A ditopological space is called
Completely semi-regular if given G ∈ SO (S) , G ⊈ Qs, there exists a semi-bi-irresolute difunction such that Ps ⊆ f←P0 and F←Q1 ⊆ G.
Completely semi-co-regular if given K ∈ SC (S) , Ps ⊈ K, there exists a semi-bi-irresolute difunction such that K ⊆ f←P0 and F←Q1 ⊆ Qs.
Completely semi-biregular if it is completely semi-regular and completely semi-co-regular.
A semi-T0 completely semi-regular (completely semi-co-regular) space is said to be semi- (respectively, semi-co-).
Proposition 6.5.A completely semi-regular space is semi-regular, and a completely semi-co-regular space is semi-co-regular.
Proof. Suppose that is completely semi-regular. Take G ∈ SO (S) and s ∈ S such that G ⊈ Qs . Let be a semi-bi-irresolute difunction such that F←Q1 ⊆ G and Ps ⊆ f←P0. Let H = F←Q1/2. Hence H ∈ SO (S) because and (f, F) is semi-irresolute. Obviously
because f←Q1 = F←Q1. In fact f←P1/2∈ SC (S) because (f, F) is semi-co-irresolute and while f←Q1/2 = F←Q1/2 = H. Therefore we can obtain Ps ⊆ H ⊆ scl (H) ⊆ G . So is semi-regular by [17, Theorem 4.8 (i)].
The other result can be shown dual to the above, and is omitted.□
The following implications hold:
Corollary 6.6.Semi-semi-T3 and semi-co-semi-co-T3.
Proposition 6.7.For a complemented ditopology , the notions of complete semi-regularity and complete semi-co-regularity properties are equivalent.
Proof. Assume that the complemented ditopology is completely semi-regular. Take K ∈ SC (S), s ∈ S, with Ps ⊈ K. By the definition of σ we have σ (K) ⊈ σ (Ps) . Choose t ∈ S satisfying σ (K) ⊈ Qt and Pt ⊈ σ (Ps). By the complemented of we have σ (K) ∈ SO (S) and there exists a semi-bi-irresolute difunction such that Pt ⊆ f←P0 and F←Q1 ⊆ σ (K). If Pt ⊈ σ (Ps) verifies Ps ⊈ σ (Pt), then σ (f←P0) ⊆ σ (Pt) . Hence we have Ps ⊈ σ (f←P0) and therefore σ (f←P0) ⊆ Qs. Otherwise K ⊆ σ (F←Q1), we get σ (f←P0) = σ (f←ι (Q1)) = (f′) ←Q1, σ (F←Q1) = σ (F←ι (P0)) = (F′) ←P0 to see that the semi-bi-irresolute difunction with K⊆ (F′) ←P0 and (f′) ←Q1 ⊆ Qs. Clearly (f, F) ′ is semi-bi-irresolute, thus is completely semi-co-regular.
Using dual arguments, it can be obtain the other direction.□
Now let us consider the preservation of complete semi-regularity and complete semi-co-regularity under initial ditopology.
Lemma 6.8.Let be a texture and be a texture for which is a chain and difunctions for k = 1, 2, . . . , n. Hence there exists a difunction for which , for all B∈ .
Proof. The mapping is defined by for B∈ . We want to show that θ preserves both arbitrary joins and arbitrary intersections. It is easy to see that by the preservation of intersections by inverse images [5, Corollary 2.26], so θ preserves arbitrary intersections. Otherwise since inverse images preserve inclusion, we have ⋁j∈Jθ (Bj) ⊆ θ (⋁ j∈JBj) for Bj∈ , j ∈ J . It remains to show that the reverse inclusion. For this, we have
by the preservation of joins by inverse images [5, Corollary 2.26] and is completely distributive. Consider α ∈ Jn. Because is a chain there exists k with 1 ≤ k ≤ n such that the finite set {Bα(k)∣ k = 1, 2, . . . , n } contains a set Bjα satisfying Bα(k)⊆ Bjα. Hence
so
as required.
The proof is completed by noting that the difunction satisfying g←B = θ (B) for all , the remaining equality being a consequence of the equality of the inverse image and inverse co-image for difunctions.□
Theorem 6.9.Let (), j ∈ J, be ditopological texture spaces and (,τj, κj) difunctions. If the ditopologies (, τj, κj) are completely semi-regular (completely semi-co-regular) for all j ∈ J, then the initial ditopology generated by the given spaces and difunctions is completely semi-regular (respectively, completely semi-co-regular).
Proof. To show that is completely semi-regular take H ∈ SO (S) and s ∈ S with H ⊈ Qs. Hence there exists Hk ∈ Sk, k = 1, 2, . . . , n such that and , therefore for k = 1, 2, . . . , n and we get sk ∈ Sk such that Hk ⊈ Qsk and . Since () is completely semi-regular there exists a semi-bi-irresolute difunction (uk, Uk) on to for which and . Let (gk, Gk) = (uk, Uk) ∘ (fk, Fk). Then for each k, and is also chain, thus by Lemma 6.8 there exists a difunction satisfying and for all . Obviously, for each k the semi-bi-irresolute of (fk, Fk) and (uk, Uk) imply that of (gk, Gk). Thus (g, G) is semi-bi-irresolute.
It remains to show that G←Q1 ⊆ H and Ps ⊆ g←P0.
If now we suppose Ps⊈ g←P0 then for some k, 1 ≤ k ≤ n, we have Ps by [5, Lemma 2.16 (2)]bed1. Therefore we have with fk⊈ and . From the definition of (fk, Fk) we now have Psk⊈ , so and we obtain the contradiction .
Thus . This completes the proof of complete semi-regularity and complete semi-co-regularity can be proved in a similar way.□
As a result, from Theorem 6.9 and the product ditopology is the initial ditopology under the projection difunctions, we have immediately:
Corollary 6.10.Let be the product of non-empty disjoint ditopological texture spaces (Sj,,τj, κj), j ∈ J . If () is completely semi-regular (completely semi-co-regular), then is completely semi-regular (respectively, completely semi-co-regular).
Definition 6.11. A ditopological space is said to be semi-normal if given F ∈ SC (S), G ∈ SO (S) with F ⊆ G there exists K ∈ SC (S), H ∈ SO (S) satisfying F ⊆ H ⊆ K ⊆ G.
A semi-T1 semi-normal space is said to be semi-T4 and a semi-co-T1 semi-normal space is said to be semi-co-T4.
Remark 6.12. Semi-normality is self-dual, thus there is no separate semi-co-normality axiom. Obviously, equivalent formulations of the semi-normality condition for are that given F ∈ SC (S), G ∈ SO (S) with F ⊆ G there exists H ∈ SO (S) such that F ⊆ H ⊆ scl (H) ⊆ G, or that there exists K ∈ SC (S) such that F ⊆ sint (K) ⊆ K ⊆ G.
Example 6.13.
Consider the ditopological texture space . If SO (S) ⊆ SC (S) or SC (S) ⊆ SO (S) then is semi-normal. In particular, (SO (S), ) and (SC (S), ) are semi-normal.
The unit interval is obviously semi-normal.
Lemma 6.14.A semi-R0 semi-normal space is semi-regular and a semi-co-R0 semi-normal space is semi-co-regular.
Proof. We prove the first result, the other result being dual. Consider G ∈ SO (S) and s ∈ S such that G ⊈ Qs. Hence scl (Ps) ⊆ G, and applying normality with F = scl (Ps) ∈ SC (S) gives H ∈ SO (S) with Ps ⊆ scl (Ps) ⊆ H ⊆ scl (H) ⊆ G. Thus is semi-regular by [17, Theorem 4.8 (i)].□
Now, we give implications as follows:
Corollary 6.15.Semi-T4⇒semi-T3 and semi-co-T4⇒semi-co-T3.
For ditopological texture spaces, we show that a counterpart of Urysohn’s Lemma enable us to strengthen this last result.
Theorem 6.16.The ditopological texture space (, τ, κ) is semi-normal iff given F ∈ SC (S), G ∈ SO (S) satisfying F ⊆ G there exists a semi-bi-irresolute difunction such that F⊆ w←P0 and W←Q1 ⊆ G.
Proof. Assume that is semi-normal. Take F ∈ SC (S), G ∈ SO (S) such that F ⊆ G. The same in the proof of the classical Urysohn’s Lemma we may inductively choose for each binary number α = k/2n, k = 2, 3, . . . , 2n - 1, , a set H (α) ∈ SO (S) satisfying
for all binary numbers α,α′ with 0 < α < α′ < 1. We now define a function by setting
where the infimum of an empty set of binary numbers is taken to be 1. If Ps ⊈ Qs′ then Ps′ ⊆ Ps, therefore Qs′ ⊆ Qs and we deduce that ω (s′) ≤ ω (s). Thus Pω(s) ⊈ Qω(s′), and we have shown that satisfies condition (a) of [5, Lemma 3.4]bed1. We deduce from this lemma that the equalities
define a difunction (w, W) : . For r∈ it follows that
We show that (3). Assume that w←Pr⊈ ⋂{scl (H (α)) ∣ r < α} and take a binary number α with r < α and w←Pr ⊈ scl (H (α)). Now, we have s ∈ S with w←Pr ⊈ Qs and Ps ⊈ scl (H (α)). From w←Pr ⊈ Qs, we have s′ ∈ S with Ps′ ⊈ Qs for which for all , whence we have ω (s) ≤ r because by (1). Thus ω (s) < α and we take α′ with α′ < α and H (α′) ⊈ Qs by the definition of ω. But this implies Ps ⊆ H (α′) ⊆ scl (H (α)), which is a contradiction.
If now we suppose ⋂ { scl (H (α)) ∣ r < α } ⊈ w←Pr then we have s ∈ S satisfying ⋂{scl (H (α)) ∣ r < α ⊈ Qs} and Ps ⊈ w←Pr. Hence we have with and Pt ⊈ Pr, that is r < t. From (1) we now have v ∈ S with Ps ⊈ Qv and . Hence t ≤ ω (v) , and so ω (v) ≤ ω (s) by condition (a), whence r < ω (s). Hence there exists α′, α with r< α′ < α < ω (s) and so r< α′ . This leads to scl (H (α′)) ⊈ Qs, and we deduce from scl (H (α′)) ⊆ H (α) that H (α) ⊈ Qs. Thus the definition of ω implies the contradiction ω (s) ≤ α. This completes the proof of (3). (3) dual to (4), and we omit the details.
Applying equality (3) now gives F⊆ w←P0 because F ⊆ H (α) ⊆ scl (H (α)) for all α, while all the elements of have the form Pr, , and from (3) we have that w←Pr ∈ SC (S). Thus (w, W) is semi-co-irresolute. Similarly (4) now gives W←Q1 ⊆ G because H (α) ⊆ G for all α, while all the elements of have the form Qr, , and from (4) we have W←Qr ∈ SO (S). Thus (w, W) is semi-irresolute.
Conversely if such that F⊆ w←P0 and W←Q1 ⊆ G then H = W←Q1/2 ∈ SO (S), K = w←P1/2 ∈ SC (S) verify F ⊆ H ⊆ K ⊆ G.
Thus the proof is completed.□
Corollary 6.17.A semi-R0 semi-normal space is completely semi-regular and a semi-co-R0 semi-normal space is completely semi-co-regular.
Proof. Let be semi-R0 and semi-normal. Take G ∈ SO (S), s ∈ S such that G ⊈ Qs. Then scl (Ps) ⊆ G, and by Theorem 6.16 with F = scl (Ps) gives a semi-bi-irresolute difunction satisfying W←Q1 ⊆ G and Ps ⊆ w←P0. Therefore is completely semi-regular. The other result can be shown dual to the above, and is omitted.□
Corollary 6.18.Semi-T4⇒semi-semi-T3 and
semi-co-T4⇒semi-co-semi-co-T3.
Proof. A semi-T4 space is semi-T1 and semi-normal, so semi-T0, semi-R0 and semi-normal, hence by Corollary 6.17 it is semi-T0 and completely semi-regular, i.e., semi-. So semi-T4⇒semi-, and semi-semi-T3 follows from Corollary 6.6. The remaining implications can be proved similarly.□
Soft fuzzy topological spaces
In this section, we recall [16] soft fuzzy topological spaces and some properties with related to ditopologies.
In particular the top and the bottom elements in the soft fuzzy lattice SF (X) are given respectively by (1, X) and (0, ∅).
Definition 7.1. A soft fuzzy topology on a non-empty set X is a family T ⊆ SF (X) of soft fuzzy sets in X satisfying the following axioms:
(0, ∅) , (1, X) ∈ T,
For any finite number of soft fuzzy sets if (μj, Mj) ∈ T, j = 1, 2, …, n then ∈T,
For any family of soft fuzzy sets if (μj, Mj) ∈ T, j ∈ J then ⊔j∈J (μj, Mj) ∈ T .
In general, the elements of T are said to be soft fuzzy open set, and those of T′ ={(μ, M) ∣ (μ, M) ′ ∈ T} soft fuzzy closed set.
The closure of a soft fuzzy set (μ, M) is denoted by and defined as follows:
Similarly the interior (μ, M) ° of (μ, M) is defined as follows:
Definition 7.2. Let T be an SF-topology on X and V an SF-topology on Y. If (v, N) ∈ V ⇒ φ← (v, N) ∈ T then a function φ : X → Y is said to be T - V continuous.
SF-topological spaces and continuous functions between the base sets define a construct category which denote by SF-Top.
Let us now relate SF-topologies on X with ditopologies on . If T is a topology on X, then T ⊆ SF (X) and we may apply the isomorphism (μ, M) → Aξ(μ,M) to give . Since Aξ(0,∅) =∅, , and the isomorphism takes meet and join in SF (X) to intersection and union respectively in , it is immediate that Aξ(T) is a topology on the plain complemented texture . On the other hand, for (μ, M) ∈ T we have Aξ((μ,M)′) = zX (Aξ(μ,M)), so Aξ(T′) = ZX (Aξ(T)) is a cotopology on . Hence (Aξ(T), Aξ(T′)) is the complemented ditopology on corresponding to T.
Remark 7.3. We observe that which is just the closure of Aξ(μ,M) with respect to the ditopology (Aξ(T), Aξ(T′)). Similarly, Aξ(μ,M)° is the interior of Aξ(μ,M).
The category whose objects are complemented ditopological texture spaces of the form (ZX,,zX, τx, κx), X ∈ Ob(Set), and whose morphisms are the bicontinuous mappings <φ, id>, φ∈Set(X, Y), is denoted by SF-Ditop. In [24], it was shown that SF-Top→SF-Ditop is an isomorphism. Hence, this isomorphism may be used to translate concepts and results for ditopological texture spaces to SF-topologies on a set X. Indeed, this will be the source of the material on semi-open, semi-closed and semi-separations presented in the next sections.
Semi-open sets in soft fuzzy topological spaces
The semi-open and semi-close sets in Soft Fuzzy Topological Spaces were definite in [16]. Now, we recall these concepts.
Definition 7.4. Let T be an SF-topology on X. Then (μ, M) ∈ SF (X) is said to be soft fuzzy semi-open (written SF-semi-open) if .
As usual, a soft fuzzy set (μ, M) ∈ SF (X) is said to be soft fuzzy semi-closed (written SF-semi-closed) if (μ, M) ′ is SF-semi-open.
On the other hand, the semi-closure of the soft fuzzy set (μ, M) is defined as follows:
Likewise, the semi-interior (μ, M) ∘S of (μ, M) is defined as follows:
Obviously, (μ, M) ∘S is the greatest SF-semi-open set which is contained in (μ, M) and is the smallest SF-semi-closed set which contains (μ, M) and we have,
Semi-separation axioms in soft fuzzy topological spaces
In Sections 5 and 6, it is given semi-separation axioms for ditopological texture spaces. We may use the isomorphism between soft fuzzy topological spaces and ditopological texture spaces to define corresponding axioms for SF-topologies. To illustrate this process we give the details for the semi-R0 and semi-co-R0 axioms.
Proposition 8.1.Let T be a SF-topology on X . Then the ditopology (Aξ(T), Aξ(T′)) on is semi-R0 iff (μ, M) SF-semi-open set,
and semi-co-R0 iff (υ, N) SF-semi-closed set,
Proof. Under the isomorphism , (μ, M) SF-semi-open set corresponds to and (xs, { x }) ∈ (μ, M) to P(x,s) ⊆ Aξ(μ,M), and then to Aξ(μ,M) ⊈ Q(x,s) because we deal with a plain texture. The semi-R0 axiom gives scl (P(x,s)) ⊆ Aξ(μ,M), which corresponds to , as required. The result for the semi-co-R0 axiom is similar and is omitted.□
Remark 8.2. By [17, Corollary 4.3], the notions of semi-R0 and semi-co-R0 coincide for complemented ditopologies. Since the ditopology (Aξ(T), Aξ(T′)) is complemented this will be the case here, we need only define the semi-R0 axiom for SF-topologies, regarding the definition of ditopological semi-co-R0 axiom as giving an alternative description of the semi-R0 axiom. Thus we have the following definition:
Definition 8.3. Let T be a SF-topology on X . Then T is said to be a soft fuzzy semi-R0-space if any one of the following equivalent conditions is true:
(μ, M) SF-semi-open set with
(υ, N) SF-semi-closed set with (xs, { x }) ∉ (υ, N) ⇒ (υ, N) ⊑ (xs, X ∖ { x }) ∘S .
Further equivalent conditions for the ditopological semi-R0 and semi-co-R0 axioms are given in [17, Theorem 4.2], and these translate easily under the mentioned isomorphism to equivalent conditions for an SF-topology to be semi-R0:
Proposition 8.4.Let T be a SF-topology on X . Then T is said to be a soft fuzzy semi-R0-space iff it satisfies any one of the following equivalent conditions:
For (μ, M) SF-semi-open set there are (υi, Ni) SF-semi-closed sets, i ∈ I, with (μ, M) = ⊔ i∈I (υi, Ni) .
Given (μ, M) SF-semi-open set with (xs, { x }) ∈ (μ, M) there exists, (υ, N) SF-semi-closed set with (υ, N) ⊑ (μ, M) and (xs, { x }) ∈ (υ, N) .
For (υ, N) SF-semi-closed set there are (μi, Mi) SF-semi-open sets, i ∈ I, with (υ, N) = ⊓ i∈I (μi, Mi) .
Given (υ, N) SF-semi-closed set with (xs, { x }) ∉ (υ, N) there exists, (μ, M) SF-semi-open set with (μ, M) ⊑ (υ, N) and (xs, { x }) ∉ (μ, M) .
Using the above treatment of the semi-R0 axiom as a guide, we now give the semi-R1 and regularity axioms without discussing the link with the corresponding ditopological axioms in detail.
Definition 8.5. Let T be a SF-topology on X . Then T is said to be a soft fuzzy semi-R1-space if any one of the following equivalent conditions is true:
(μ, M) SF-semi-open set with (xs, { x }) ∈ (μ, M) and (yt, { y }) ∉ (μ, M) ⇒ ∃ (υ, N) SF-semi-open set with (xs, { x }) ∈ (υ, N) and
(υ, N) SF-semi-closed set with (xs, { x }) ∉ (υ, N) and (yt, { y }) ∈ (υ, N) ⇒ ∃ (μ, M) SF-semi-closed set with (xs, { x }) ∉ (μ, M) and (yt, { y }) ∈ (μ, M) ∘S .
From [17, Theorem 4.5] we get the further equivalent conditions given below.
Proposition 8.6.Let T be a SF-topology on X . Then T is said to be a soft fuzzy semi-R1-space iff it satisfies any one of the following equivalent conditions:
For (μ, M) SF-semi-open set we have SF-semi-open sets, i ∈ I, j ∈ Ji, with
Given (μ, M) SF-semi-open set with (xs, { x }) ∈ (μ, M) and (yt, { y }) ∉ (μ, M) , then we have (υ, N) SF-semi-open set with (xs, { x }) ∈ (υ, N)
For (υ, N) SF-semi-closed set we have SF-semi-closed sets, i ∈ I, j ∈ Ji, with
Given (υ, N) SF-semi-closed set with (xs, { x }) ∉ (υ, N) and (yt, { y }) ∈ (υ, N) , then we have (μ, M) SF-semi-closed set with (yt, { y }) ∈ (μ, M) ∘S ⊑ (υ, N) ⊑ (xs, X ∖ { x }) .
Definition 8.7. Let T be a SF-topology on X . Then T is said to be a soft fuzzy semi-regular space if any one of the following equivalent conditions is true:
(μ, M) SF-semi-open set with (xs, { x }) ∈ (μ, M) ⇒ ∃ (υ, N) SF-semi-open set with (xs, { x }) ∈ (υ, N) and
(υ, N) SF-semi-closed set with (xs, { x }) ∉ (υ, N) ⇒ ∃ (μ, M) SF-semi-closed set with (xs, { x }) ∉ (μ, M) and (υ, N) ⊑ (μ, M) ∘S .
According to [17, Theorem 4.8] we have:
Proposition 8.8.Let T be a SF-topology on X . Then T is said to be a soft fuzzy semi-regular space iff it satisfies any one of the following equivalent conditions:
For (μ, M) SF-semi-open set we have (μi, Mi) SF-semi-open sets, i ∈ I, with
For (υ, N) SF-semi-closed set we have (υi, Ni) SF-semi-closed sets, i ∈ I, with (υ, N) = ⊓ i∈I (υi, Ni) = ⊓ i∈I (υi, Ni) ∘S .
It is obvious from the definitions that soft fuzzy semi-regular ⇒ soft fuzzy semi-R1⇒ soft fuzzy semi-R0.
Coming back to the semi-T0 axiom, which is a self-dual condition of ditopological texture spaces. In Section 5, it was given that many equivalent conditions for a ditopological texture space to be semi-T0. Now, we give the semi-T0 axiom for SF-topological spaces.
Definition 8.9. Let T be a SF-topology on X . Then T is said to be a soft fuzzy semi-T0-space if any one of the following equivalent conditions is true:
(xs, { x }) ∉ (yt, { y }) ⇒ ∃ (μi, Mi)∈(SC (S) ∪ SO (S)), i ∈ I satisfying (yt, { y }) ∈ ⊔ i∈I(μi, Mi) ⊑ (xs, X ∖ { x }) .
(xs, { x }) ∉ (yt, { y }) ⇒ ∃ (μi, Mi)∈(SC (S) ∪ SO (S)), i ∈ I satisfying (yt, { y }) ∈ ⊓ i∈I(μi, Mi) ⊑ (xs, X ∖ { x }) .
(xs, { x }) ∉ (yt, { y }) ⇒ ∃ (μ, M)∈(SC (S) ∪ SO (S)) satisfying (xs, { x }) ∉ (μ, M) and (yt, { y }) ∈ (μ, M) .
and (xs, X ∖ { x }) ∘S ⊑ (yt, X ∖ { y }) ∘S ⇒ (xs, X ∖ { x }) ∈ (yt, X ∖ { y }) .
Thus, we have:
Definition 8.10. Let T be a SF-topology on X . Then T is called:
soft fuzzy semi-T1-space if it is soft fuzzy semi-T0 and soft fuzzy semi-R0 .
soft fuzzy semi-T2-space if it is soft fuzzy semi-T0 and soft fuzzy semi-R1 .
soft fuzzy semi-T3-space if it is soft fuzzy semi-T0 and soft fuzzy semi-regular.
Obviously, soft fuzzy semi-T3⇒soft fuzzy semi-T2⇒soft fuzzy semi-T1⇒soft fuzzy semi-T0 the same as for classical topology. By Theorem 5.6 we get the following characterizations of the soft fuzzy semi-T1 property.
Proposition 8.11.ASF-topology T on X is soft fuzzy semi-T1-space iff it satisfies any one of the following equivalent conditions:
For (υ, N) ∈ SF (X) we have (υi, Ni) SF-semi-closed sets, i ∈ I, with (υ, N) = ⊔ i∈I (υi, Ni) .
(xs, { x }) ∉ (yt, { y }) ⇒ ∃ (υ, N) SF-semi-closed set satisfying (xs, { x }) ∉ (υ, N) ∉ (yt, X ∖ { y }) .
For any (μ, M) ∈ SF (X) we have (μi, Mi) SF-semi-open sets, i ∈ I, with (μ, M) = ⊓ i∈I (μi, Mi) .
(xs, { x }) ∉ (yt, { y }) ⇒ ∃ (μ, M) SF-semi-open set satisfying (xs, { x }) ∉ (μ, M) ∉ (yt, X ∖ { y }) .
Finally, by Theorem 5.9 we get the following characterizations of the soft fuzzy semi-T2 for SF-topological spaces.
Proposition 8.12.ASF-topology T on X is soft fuzzy semi-T2-space iff it satisfies any one of the following equivalent conditions:
(xs, { x }) ∉ (yt, { y }) ⇒ ∃ (μ, M) SF-semi-open set and ∃ (υ, N) SF-semi-closed set with (μ, M) ⊑ (υ, N) , (xs, { x }) ∉ (υ, N) and (yt, { y }) ∈ (μ, M) .
For (μ, M) ∈ SF (X) there exist SF-semi-open sets and SF-semi-closed sets, i ∈ I, j ∈ Ji, with for all i, j so that
Definition 8.13. A SF-topology T on X is soft fuzzy completely semi-regular if given (μ, M) SF-semi-open set with (xs, { x }) ∈ (μ, M) there exists a inverse image satisfying (xs, { x }) ⊑ φ← (0s, { 0 }) and .
Finally we note the following:
Definition 8.14. A SF-topology T on X is soft fuzzy semi normal if given (μ, M) SF-semi-open set and (υ, N) SF-semi-closed set with (υ, N) ⊑ (μ, M) there exists (φ, G) SF-semi-open set satisfying
Conclusion
The theory of textures is a point-set based setting for fuzzy sets. Many properties of Hutton algebras can be discussed in terms of textures. Furthermore, ditopologies on textures unify the classical topologies, fuzzy topologies and bitopologies in a non-complemented setting by means of duality in textural concepts. Recent works on textures show that they also proved useful model for soft fuzzy topological spaces. In the Section 4, it is proved that there is the categorical isomorphism between textures and soft fuzzy sets. Further the category of soft fuzzy topological spaces is isomorphic to the category SF-Ditop of complemented ditopological texture spaces.
In this paper, we introduce the exposition of the semi-separation axioms in ditopological texture spaces by using the notions of semi-open and semi-closed sets. We have also established the properties of these concepts and the relations between them. Further, we prove that semi-separations axioms are productive in ditopological texture spaces, and the productivity is obtained automatically in soft fuzzy topological spaces by using the mentioned categorical isomorphism. Consequently, this categorical isomorphism leads to the further study for soft fuzzy topological spaces.
On the other hand, categorical property of semi-separation axioms in soft fuzzy topological spaces would be study. Also, in this present work we restrict our interest to elementary the point semi-separation axioms and further semi-regularity axioms of textures, and leave the problem of Urysohn metrization theorem of advanced subjects in soft fuzzy topological spaces to future considerations.
Footnotes
Acknowledgements
The authors would like to thank the referees and editors for their helpful comments that have helped improve the presentation of this paper.
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