A textural view of semi-separation axioms in soft fuzzy topological spaces

Abstract

In the present paper, we have continued to study the properties of semi-separation axioms in ditopological texture spaces. We prove that the semi-T_i-spaces ( $i = 0, 1, 2, 3, 3 \frac{1}{2}, 4$ ) 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.

Keywords

Ditopology rough sets soft fuzzy sets semi-separation axioms

1 Introduction

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 $𝕀$ 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 $S$ is a family of subsets of a given universe S satisfying certain conditions which are related to the properties of the power set $P (S)$ . Then the pair $(S, S)$ is called a texture space.

It is studied T_i-spaces ( $i = 0, 1, 2, 3, 3 \frac{1}{2}, 4$ ), 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 $𝕀 = [0, 1]$ -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 $(X, P (X))$ and the unit interval texture $(𝕀, I)$ where $I = {[0, t] ∣ t \in [0, 1]} \cup {[0, t) ∣ t \in [0, 1]}$ . 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 $(X, P (X)) \otimes (𝕀, I)$ 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.

2 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 $S$ is textured by S, then we call $(S, S)$ a texture space or shortly, texture.

For a texture $(S, S)$ , most properties are appropriately defined in terms of the p-sets $P_{s} = ⋂ {A \in S ∣ s \in A}$ and, as a dually, the q-sets, $Q_{s} = ⋁ {A \in S ∣ s \notin A}$ .

Complementation: In general, a texturing of S need not be closed under set complementation, however there exists a mapping $σ : S \to S$ is said to be a complementation if σ (σ (A)) = A, $\forall A \in S$ and A ⊆ B implies σ (B) ⊆ σ (A) for $\forall A, B \in S$ . A texture with a complementation is called a complemented texture.

Example 2.1. Some examples of texture spaces,

The pair $(X, P (X))$ is called the discrete texture on X where $P (X)$ is the power set of X. Obviously, for all x ∈ X we have P_x = {x}, Q_x = X \ {x} and $π : P (X) \to P (X)$ is the ordinary complementation on $(X, P (X))$ defined by π (Y) = X \ Y for all Y ⊆ X. Therefore, $(X, P (X))$ is plain.

Let L = (0, 1], $L = {(0, r] ∣ r \in [0, 1]}$ and λ ((0, r]) = (0, 1 - r], r ∈ [0, 1]. Then $(L, L, λ)$ is complemented texture space. Here P_r = Q_r = (0, r] for all r ∈ L. Hence, $(L, L)$ is not plain.

For $𝕀 = [0, 1]$ define $I = {[0, t] ∣ t \in [0, 1]} \cup {[0, t) ∣ t \in [0, 1]}$ , ι ([0, t]) = [0, 1 - t) and ι ([0, t)) = [0, 1 - t], t ∈ [0, 1]. $(𝕀, I, ι, τ_{𝕀}, κ_{𝕀})$ is a complemented texture, which we refer to as the unit interval texture. Here P_t = [0, t] and Q_t = [0, t) for all $t \in 𝕀$ . The natural ditopology is given by $τ_{𝕀} = {[0, t) ∣ t \in [0, 1]} \cup {𝕀}$ and $κ_{𝕀} = {[0, t] ∣ t \in [0, 1]} \cup {\emptyset}$ .

If $(S, S)$ , $(T, T)$ are textures, the product texturing $S \otimes T$ of S × T consists of arbitrary intersections of sets of the form (A × T) ∪ (S × B), $A \in S, B \in T$ , and $(S \times T, S \otimes T)$ is called the product of $(S, S)$ and $(T, T)$ . For s ∈ S, t ∈ T we obviously have P_(s,t) = P_s × P_t and Q_(s,t) = (Q_s × T) ∪ (S × Q_t).

Ditopological Texture Space: A ditopology on a texture $(S, S)$ is a pair (τ, κ) of subsets of $S$ , where the set of open setsτ and the set of closed setsκ verifies $\begin{matrix} S, \emptyset \in τ and S, \emptyset \in κ . \\ G_{1}, G_{2} \in τ \Rightarrow G_{1} \cap G_{2} \in τ, and \\ K_{1}, K_{2} \in κ \Rightarrow K_{1} \cup K_{2} \in κ, \\ G_{i} \in τ, i \in I \Rightarrow ⋁_{i} G_{i} \in τ, and \\ K_{i} \in κ, i \in I \Rightarrow ⋂_{i} K_{i} \in κ . \end{matrix}$

Therefore a ditopology is essentially a “topology” for which there is no a priori relation between the open and closed sets.

For $A \in S$ we define the closurecl (A) and the interiorint (A) of A under (τ, κ) by the equalities $\begin{matrix} cl (A) & = & ⋂ {K \in κ ∣ A \subseteq K}, \\ int (A) & = & ⋁ {G \in τ ∣ G \subseteq A} . \end{matrix}$

If (τ, κ) is a ditopology on a complemented texture $(S, S, σ)$ 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 $(S_{j}, S_{j})$ , j ∈ J, are textures, $S = \prod_{j \in J} S_{j}$ and $A_{k} \in S_{k}$ for some k ∈ J we write $E (k, A_{k}) = \prod_{j \in J} Y_{j} where Y_{j} = {\begin{matrix} A_{j}, if j = k \\ S_{j}, otherwise . \end{matrix}$ Then the product texturing $S = ⨂_{j \in J} S_{j}$ on S consists of arbitrary intersections of elements of the set $E = {⋃_{j \in J_{1}} E (j, A_{j}) ∣ J_{1} \subseteq J, A_{j} \in S_{j} for j \in J_{1}} .$

Product Ditopological Texture Spaces: Let $(S_{j}, S_{j})_{j \in J}$ be ditopological texture spaces and $(S, S)$ the product of the textures $(S_{j}, S_{j})_{j \in J}$ . For each j ∈ J, $\begin{matrix} π_{j} & = & ⋁ {{\bar{P}}_{(s, s_{j})} ∣ s = (s_{k}) \in S}, \\ Π_{j} & = & ⋂ {{\bar{Q}}_{(s, s_{j})} ∣ s = (s_{k}) \in S^{♭}} \end{matrix}$ define the j-th projection difunction (π_j, Π_j) on $(S,_{j}^{\leftarrow} K = E (j, K) ∣ K \in κ_{j}, j \in J}$ is called the product ditopology on $(S, S)$ [6]. Note that the product ditopology is the coarsest ditopology making the projection difunctions (π_j, Π_j), j ∈ J are bicontinuous.

Sum Ditopological Texture Spaces: Let $(S_{i}, S_{i}),$ i ∈ I be textures with S_i∩ S_j = ∅ for i ≠ j. Let S = ⋃ _i∈IS_i and $S = {A ∣ A \subseteq S, A \cap S_{i}, \forall i \in I}$ . Then $(S, S)$ is a texture which is called sum of disjoint textures $(S_{i}, S_{i}),$ i ∈ I and if $(S_{i}, S_{i}, σ_{i})$ are the complemented textures for all i ∈ I then the complementation $σ : S \to S, σ (A) \cap S_{i} = σ_{i} (A \cap S_{i}), i \in I$ makes $(S, S, σ)$ is a complemented texture.

We have noted that by [6]; If $(S, S, σ)$ is a complemented sum texture of $(S_{i}, S_{i}, σ_{i})$ , then for j ∈ I it can be obtained the following equalities:

P_s = P_{s
_j} × { j } ,

Q_s = (Q_{s
_j} × { j }) ∪ (⋃ _k∈I∖{j}S_k × { k }) .

Direlation: Let us consider the product texture $P (S) \otimes T$ of the texture spaces $(S, P (S))$ and $(T, T)$ and denote the p-sets and the q-sets by ${\bar{P}}_{(s, t)}$ and ${\bar{Q}}_{(s, t)}$ , respectively. Obviously, ${\bar{P}}_{(s, t)} = {s} \times P_{t}$ and ${\bar{Q}}_{(s, t)} = (S \ {s} \times T) \cup (S \times Q_{t})$ where t ∈ T and s ∈ S. Hence

$r \in P (S) \otimes T$ is said to be a relation from $(S, S)$ to $(T, T)$ if it satisfies

$r ⊈ {\bar{Q}}_{(s, t)}, P_{s^{'}} ⊈ Q_{s} \Rightarrow r ⊈ {\bar{Q}}_{(s^{'}, t)}$ .

$r ⊈ {\bar{Q}}_{(s, t)} \Rightarrow \exists s^{'} \in S$ satisfying P_s ⊈ Q_s′ and $r ⊈ {\bar{Q}}_{(s^{'}, t)}$ .

$R \in P (S) \otimes T$ is said to be a corelation from $(S, S)$ to $(T, T)$ if it satisfies

${\bar{P}}_{(s, t)} ⊈ R, P_{s} ⊈ Q_{s^{'}} \Rightarrow {\bar{P}}_{(s^{'}, t)} ⊈ R$ .

${\bar{P}}_{(s, t)} ⊈ R \Rightarrow \exists s^{'} \in S$ satisfying P_s′ ⊈ Q_s and ${\bar{P}}_{(s^{'}, t)} ⊈ R$ .

A pair (r, R), where r is a relation and R a corelation from $(S, S)$ to $(T, T)$ , is said to be a direlation from $(S, S)$ to $(T, T)$ .

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 $(S, S)$ to $(T, T)$ . Then (f, F) is said to be a difunction from $(S, S)$ to $(T, T)$ if it satisfies the following two conditions.

For s, s′ ∈ S, P_s ⊈ Q_s′⇒ ∃ t ∈ T with $f ⊈ {\bar{Q}}_{(s, t)}$ and ${\bar{P}}_{(s^{'}, t)} ⊈ F$ .

For t, t′ ∈ T and s ∈ S, $f ⊈ {\bar{Q}}_{(s, t)}$ and ${\bar{P}}_{(s, t^{'})} ⊈ F \Rightarrow P_{t^{'}} ⊈ Q_{t}$ .

Definition 2.2. Let $(f, F) : (S, S) \to (T, T)$ be a difunction. For $A \in S$ and $B \in T$ , the A-sections and the B-presections with respect to (f, F) are given as follows: $\begin{matrix} f^{\to} A & = & ⋂ {Q_{t} ∣ \forall s, f ⊈ {\bar{Q}}_{(s, t)} \Rightarrow A \subseteq Q_{s}}, \\ F^{\to} A & = & ⋁ {P_{t} ∣ \forall s, {\bar{P}}_{(s, t)} ⊈ F \Rightarrow P_{s} \subseteq A}, \\ and \\ f^{\leftarrow} B & = & ⋁ {P_{s} ∣ \forall t, f ⊈ {\bar{Q}}_{(s, t)} \Rightarrow P_{t} \subseteq B}, \\ F^{\leftarrow} B & = & ⋂ {Q_{s} ∣ \forall t, {\bar{P}}_{(s, t)} ⊈ F \Rightarrow B \subseteq Q_{t}}, \end{matrix}$ 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 $(f, F) : (S, S) \to (T, T)$ be a difunction. Then (f, F) is said to be surjective if it satisfies the condition

SUR. For t, t′ ∈ T, P_t ⊈ Q_t′⇒ ∃ s ∈ S with $f ⊈ {\bar{Q}}_{(s, t^{'})}$ and ${\bar{P}}_{(s, t)} ⊈ F$ .

Likewise, (f, F) is said to be injective if it satisfies the condition

INJ. For s, s′ ∈ S and t ∈ T, $f ⊈ {\bar{Q}}_{(s, t)}$ and ${\bar{P}}_{(s^{'}, t)} ⊈ F \Rightarrow P_{s} ⊈ Q_{s^{'}}$ .

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 $(S, S),$ $(T, T)$ are textures and φ : S → T an ω-compatible point mapping, namely one satisfying that if P_s ⊈ Q_s′ then P_φ(s) ⊈ Q_φ(s′), hence the equalities $\begin{matrix} f_{φ} & = & ⋁ {{\bar{P}}_{(s, t)} ∣ \exists u \in S \Rightarrow P_{φ (u)} ⊈ Q_{t}, P_{s} ⊈ Q_{u}}, \\ F_{φ} & = & ⋂ {{\bar{Q}}_{(s, t)} ∣ \exists u \in S \Rightarrow P_{t} ⊈ Q_{φ (u)}, P_{u} ⊈ Q_{s}}, \end{matrix}$ define a difunction (f_φ, F_φ) on $(S, S)$ to $(T, T)$ . Moreover, for $B \in T$ we have $F_{φ}^{\leftarrow} B = φ^{\leftarrow} B = f_{φ}^{\leftarrow} B$ , where φ^←B = ⋁ {P_s ∣ P_φ(u) ⊆ B, ∀u ∈ S with P_s ⊈ Q_u}.

We will denote by ifTex the construct of textures and ω-preserving mappings between them.

2.1 Semi-open sets and semi-closed sets

Definition 2.5. Let $(S, S, τ, κ)$ be ditopological space. Then $A \in S$ 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 $(S, S, τ, κ)$ be a ditopological space and A∈ $S$ . We define

The semi-interior sint (A) of A is the set $sint (A) = ⋁ {B ∣ B is semi - open and B \subseteq A},$

The semi-closure scl (A) of A is the set $scl (A) = ⋂ {B ∣ B is semi - closed and A \subseteq B} .$

Obviously, for a complemented ditopological texture space $(S, S, σ, τ, κ)$ : $A \in S$ is semi-open iff σ (A) is semi-closed.

Definition 2.7. [14] Let $(S_{j}, S_{j}, τ_{j}, κ_{j}),$ j = 1, 2 be ditopological spaces. A difunction $(f, F) : (S_{1}, S_{1}) \to (S_{2}, S_{2})$ is said to be

semi-continuous (semi-irresolute) if for each open (semi-open) set $A \in S_{2}$ , the inverse image F^← (A) $\in S_{1}$ is a semi-open set.

semi-cocontinuous (semi-co-irresolute) if for each closed (semi-closed) set $B \in S_{2}$ , the inverse image f^← (B) $\in S_{1}$ 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].

2.2 Semi-regularity axioms

Definition 2.8. Let $(S, S, τ, κ)$ be a ditopological texture space. Then (τ, κ) is called

semi-R₀ if A ∈ SO (S) , A ⊈ Q_s⇒scl (P_s) ⊆ A, and

semi-co-R₀ if B ∈ SC (S) , P_s ⊈ B⇒B ⊆ sint (Q_s).

Definition 2.9. Let $(S, S, τ, κ)$ be a ditopological texture space. Then (τ, κ) is called

semi-R₁ if A ∈ SO (S) , A ⊈ Q_s, P_t ⊈ A⇒ ∃ G ∈ SO (S) , G ⊈ Q_s, P_t ⊈ scl (G), and

semi-co-R₁ if B ∈ SC (S) , P_s ⊈ B, B ⊈ Q_t⇒ ∃ K ∈ SC (S) , P_s ⊈ K, sint (K) ⊈ Q_t.

Definition 2.10. Let $(S, S, τ, κ)$ be a ditopological texture space. Then (τ, κ) is called

semi-regular if A ∈ SO (S) , A ⊈ Q_s⇒ ∃ G ∈ SO (S) , G ⊈ Q_s, scl (G) ⊆ A, and

semi-co-regular if B ∈ SC (S) , P_s ⊈ B⇒ ∃ K ∈ SC (S) , P_s ⊈ K, B ⊆ sint (K).

3 An overview of soft topological spaces

Definition 3.1. [21] Let X be an universe and E be a set of parameters. Let $P (X)$ 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 $F : A \to P (X)$ .

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 $(F, A) = {(e, F (e)) ∣ e \in E, F (e) \in P (X)} .$ A soft set (F, A) over X is said to be

a null soft set and denoted by $\tilde{\emptyset},$ if F (e) = ∅ , for each e ∈ A .

an absolute soft set and denoted by $\tilde{A}$ , 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:

$\tilde{\emptyset},$ $\tilde{X}$ ∈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 $𝕀^{X}$ denotes the family of all fuzzy sets on X where $𝕀 = [0, 1) .$ 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 $𝕀^{X},$ i.e., $μ : E \to 𝕀^{X}$ such that μ (e) ≠ 0_X, if e ∈ A and μ (e) = 0_X, otherwise, where 0_X denotes the empty fuzzy set on X .

Then a fuzzy soft set (μ, A) is defined as follows: $(μ, A) = {(a, {x_{μ_{a} (x)} ∣ x \in X}) ∣ a \in A}$ where the symbol “a ={ x_{μ_a(x)} ∣ x ∈ X }” indicates that the membership degree of the element x ∈ X is μ_a (x) where μ_a : A→ $𝕀$ is the membership function of the fuzzy set μ (a) .

$F (X, E)$ denotes the family of fuzzy soft sets over X . Then

$(μ, E) \in F (X, E)$ is said to be a null fuzzy soft set and denoted by $\tilde{\emptyset},$ if μ (e) = ∅ _X for each e ∈ E .

$(μ, E) \in F (X, E)$ is said to be an absolute fuzzy soft set and denoted by $\tilde{1_{E}}$ , if μ (e) =1_X 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:

$\tilde{\emptyset},$ $\tilde{1_{E}}$ ∈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 $𝕀 -$ fuzzy subsets of X is isomorphic to texturing $P (X) \otimes L$ where $L = {(0, r] ∣ r \in (0, 1]} .$ Now let $I = {[0, r],$ [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 $P (X) \otimes I$ can be presented as pairs (μ, M) , where μ ∈ F (X) and $M \in P (X) .$ 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 $𝕀 -$ fuzzy sets. Moreover, soft fuzzy topologies on X are specialized the notion of $𝕃 -$ topologies on X . Then we have a considerable simplification arising from the very clean point structure.

4 Lattice of soft fuzzy sets

We begin by recalling [25] that if (N, ≤) is a poset (i.e. partially ordered set) then the set $L_{N} = {L \subseteq N ∣ n \in L, m \leq n \Rightarrow m \in L}$ 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 $L_{X \times N}$ denotes the corresponding texturing of X × N, and according to [25, Proposition 4.1]t1 the texture $(X \times N, L_{X \times N})$ is the product of the texture spaces $(N, L_{N})$ and $(X, L_{X})$ . The elements of $L_{X \times N}$ may be regarded as $L_{N}$ -fuzzy subsets of X. Indeed, if we set $F_{N} (X) = {μ : X \to L_{N} ∣ x \leq y \Rightarrow μ (y) \subseteq μ (x)}$ then by [25, Proposition 4.2]t1 the mapping $μ \to A_{μ}, A_{μ} = {(x, n) ∣ n \in μ (x)}$ is an isomorphism from F_N (X) to the texturing $L_{X \times N}$ of U × N such that inverse A → μ (A), μ_A (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 (P_n) = Q_n′, n ∈ N, and conversely [25, Theorem 2.10]t1. Note that μ ∈ F_N (X) define μ′ (x) = { n ∈ N ∣ n′ ∉ μ (x′) } . Hence the mapping μ → μ′ is an order-reversing involution on F_N (X). Also (x, n) → (x, n) ′ = (x′, n′) is an order-reversing involution on (X × N, ≤), and the corresponding complementation σ_X×N on $(X \times N, L_{X \times N})$ is the product σ_X ⊗ σ_N of σ_X and σ_N . For μ∈ F_N (X), we have A_μ ={ (x, n) ∣ x ∈ U, n ∈ N, n′ ∈ μ (x′) } and σ_U×N (A_μ) = A_μ′ .

As a result, the mapping μ → A_μ becomes an isomorphism between the Hutton algebras F_N (X) and $L_{X \times N}$ .

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 $(X, P (X), π_{X})$ which is given Examples 2.1 (i). Secondly, we consider $𝕀 = [0, 1]$ with the usual ordering and the order reversing involution t → 1 - t, $t \in 𝕀$ . Then, the corresponding texture is unit interval texture $(𝕀, I, ι)$ which is given in Examples 2.1 (iii). This gives us the family $F_{𝕀}$ of $L_{I}$ -fuzzy subsets of X, and the textural representation $(X, P (X), π_{X}) \otimes (𝕀, I, ι)$ . Here, since X has discrete ordering we just have $F_{𝕀} (X) = {η ∣ η : X \to I}$ . Then for x ∈ X we have η (x) = P_t or η (x) = Q_t for some $t \in 𝕀$ .

Note that the textural presentation of the Hutton algebra F (X) of classical fuzzy sets (that is $𝕀$ -fuzzy subsets of X) is the product texture $(X, P (X), π_{X}) \otimes (L, L, λ)$ where $(L, L, λ)$ is given in Examples 2.1 (ii).

In [24], the element of $F_{𝕀} (X)$ is represent as pairs (μ, M), where $M \in P (X)$ 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 $𝕀$ -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 $F_{𝕀} (X)$ to SF (X) is given by η → (η₁, η₂) where $η_{1} (x) = sup η (x), η_{2} (x) = {x \in X ∣ η_{1} (x) \in η (x)} .$ Conversely if (μ, M) ∈ SF (X) then we may set ξ (μ, M) = η where $η (x) = {\begin{matrix} P_{μ (x)}, x \in M \\ Q_{μ (x)}, x \notin M \end{matrix}$

Lemma 4.2.The mapping ξ : SF (X) → $F_{𝕀} (X)$ defined above is a bijection with inverse ξ^-1 given by ξ^-1 (η) = (η₁, η₂).

A simple calculation shows that $A_{ξ (μ, M)} = {(x, s) ∣ s < η (x) or (s = η (x), x \in M)} .$

Definition 4.3. The relation ⊑ on SF (X) is given by (μ, M) ⊑ (v, N) ⇔ A_ξ(μ,M) ⊆ A_ξ(v,N).

We have an order-preserving bijection between (SF (X) , ⊑) and $(P (X) \otimes I, \subseteq)$ because inclusion is a partial order on $P (X) \otimes I$ .

Proposition 4.4. Let $F = {(μ_{j}, M_{j}) \in SF (X) ∣ j \in J} .$

$F$ has a meet in (SF (X) , ⊑), denoted by ⊓_j∈J(μ_j, M_j) and considered as $⊓_{j \in J} (μ_{j}, M_{j}) = (μ, M),$ where μ (x) = ⋀ _j∈Jμ_j (x) and $M = {x \in X ∣ \forall j \in J, x \in M_{j} or μ (x) < μ_{j} (x)}$ for all x ∈ X.

$F$ has a join in (SF (X) , ⊑), denoted by ⊔_j∈J(μ_j, M_j) and considered as $⊔_{j \in J} (μ_{j}, M_{j}) = (μ, M),$ where μ (x) = ⋁ _j∈Jμ_j (x) and $M = {x \in X ∣ \exists j \in J with x \in M_{j}, μ (x) = μ_{j} (x)}$ for all x ∈ X.

Corollary 4.5.For all (μ_j, M_j) ∈ SF (X) , j ∈ J, we have $\begin{matrix} A_{ξ (⊓_{j \in J} (μ_{j}, M_{j}))} & = & ⋂_{j \in J} A_{ξ (μ_{j}, M_{j})}, and \\ A_{ξ (⊔_{j \in J} (μ_{j}, M_{j}))} & = & ⋃_{j \in J} A_{ξ (μ_{j}, M_{j})} . \end{matrix}$ (SF (X) , ⊑) is a complete lattice since $(P (X) \otimes I, \subseteq)$ is a complete lattice. In particular, ξ is an isomorphism between (SF (X) , ⊑) and $F_{𝕀} (X)$ .

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 $F_{𝕀} (X)$ , and hence to $L_{X \times I} = P (X) \otimes I$ .

In this study, the product texture space $(X, P (X), π_{X})$ $\otimes (𝕀, I, ι)$ will denoted by $(Z_{X}, Z_{X}, z_{X})$ , that is $\begin{matrix} (X, P (X), π_{X}) \otimes (𝕀, I, ι) \\ = (X \times 𝕀, P (X) \otimes I, π_{X} \otimes ι) = (Z_{X}, Z_{X}, z_{X}) . \end{matrix}$ By the definition of π_X ⊗ ι, $z_{X} A_{ξ (μ, M)} = A_{ξ (μ, M)^{'}} = A_{ξ (1 - μ, X ∖ M)}$ whenever (μ, M) ∈ SF (X) .

Now consider the soft fuzzy points (x_s, { x }) and the soft fuzzy copoint (x^s, X ∖ { x }) in SF (X) defined by $x_{s} (z) = {\begin{matrix} s, if z = x \\ 0, otherwise \end{matrix} and x^{s} (z) = {\begin{matrix} s, if z = x \\ 0, otherwise \end{matrix},$ where base x ∈ X and value $s \in 𝕀 .$ We denote (x_s, { x }) ⊑ (μ, M) by (x_s, { x }) ∈ (μ, M) , and refer to (x_s, { x }) as an element of (μ, M) .

4.1 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 > : Z_X → Z_Y defined by <φ, id > (x, s) = (φ (x) , s) is ω-preserving which is given in Remark 2.4, regarded as a mapping from $(Z_{X}, Z_{X})$ to $(Z_{Y}, Z_{Y}),$ where id is the identity on $𝕀$ . Let us denote by SF-Set the construct whose objects are pairs (X, SF (X) ) and morphisms from Set. Specially we define $G :$ SF-Set→ifTex by setting $\begin{matrix} G ((X, SF (X)) \overset{φ}{\to} (Y, SF (Y))) \\ = (Z_{X}, Z_{X}) \overset{< φ, id >}{\to} (Z_{Y}, Z_{Y}) . \end{matrix}$ 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 $\begin{matrix} f & = & ⋃ {{\bar{P}}_{((x, s), (φ (x), s))} ∣ (x, s) \in X \times 𝕀}, \\ F & = & ⋂ {{\bar{Q}}_{((x, s), (φ (x), s))} ∣ (x, s) \in X \times 𝕀} . \end{matrix}$ The image and co-image operators map from $Z_{X}$ to $Z_{Y}$ , and inverse image and inverse co-image operators, which are equal, map from $Z_{Y}$ to $Z_{X}$ . In view of the isomorphism between SF (X) and $Z_{X}$ , and that between SF (Y) and $Z_{Y}$ , 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 $φ^{\leftarrow} (v, N) = (v \circ φ, φ^{- 1} [N]) .$

5 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 $(S, S, τ, κ)$ be ditopological space. If $A \subseteq S$ contain S, ∅ and be closed under arbitrary joins. Then the following are equivalent:

A = ⋂ _j∈JA_j for all $A \in S$ and $\exists A_{j} \in A$ ,j ∈ J .

For all s, t ∈ S, if P_s ⊈ P_t then there exists $A \in A$ satisfying P_t ⊆ A ⊆ Q_s .

For all s, t ∈ S, if Q_s ⊈ Q_t then there exists $A \in A$ satisfying P_t ⊆ A ⊆ Q_s .

Proof. (i) ⇒ (ii) Take P_t = A . For some $A_{j} \in A$ ,j ∈ J we have P_s ⊈ P_t = ⋂ _j∈JA_j . Therefore, for some j ∈ J, A_j ⊆ Q_s and P_t ⊆ A_j .

(ii) ⇒ (iii) Consider Q_s ⊈ Q_t. According to definition, Q_s =⋁ { P_u ∣ P_s ⊈ P_u } so we have u ∈ S with P_s ⊈ P_u and P_u ⊆ Q_s. We can write P_u = A, thus we have P_t ⊆ A, whence P_t ⊆ A ⊆ Q_s.

(iii) ⇒ (i) Consider $Q_{s} \in A$ , hence we obtain A =⋂ { Q_s ∣ s ∉ A }, for all $A \in S$ . □

Theorem 5.2.Let $(S, S, τ, κ)$ be ditopological space. If $B \subseteq S$ contain S, ∅ and be closed under arbitrary intersections. Then the following are equivalent:

B = ⋁ _j∈JB_j for all $B \in S$ and $\exists B_{j} \in B$ , j ∈ J .

For s, t ∈ S, if Q_s ⊈ Q_t then there exists $B \in B$ satisfying P_s ⊈ B ⊈ Q_t .

For s, t ∈ S, if P_s ⊈ P_t then there exists $B \in B$ satisfying P_s ⊈ B ⊈ Q_t .

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

$A = {(SC (S) \cup SO (S))}^{\lor}$ in Theorem 5.1, and

$B = {(SC (S) \cup SO (S))}^{\cap}$ 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:

P_s ⊈ P_t ⇒ ∃ C_j ∈ (SC (S) ∪ SO (S)), j ∈ J satisfying P_t ⊆ ⋁ _j∈JC_j ⊆ Q_s .

Q_s ⊈ Q_t ⇒ ∃ C_j ∈ (SC (S) ∪ SO (S)), j ∈ J satisfying P_t ⊆ ⋂ _j∈JC_j ⊆ Q_s .

Q_s ⊈ Q_t ⇒ ∃ C ∈ (SC (S) ∪ SO (S)) satisfying P_s ⊈ C ⊈ Q_t .

scl (P_s) ⊆ scl (P_t) and sint (Q_s) ⊆ sint (Q_t) ⇒ Q_s ⊆ Q_t .

Proof. We need only verify (iv) since the other conditions follow from Theorem 5.1 with

$A = {(SC (S) \cup SO (S))}^{\lor}$ , or from Theorem 5.2 with $B = {(SC (S) \cup SO (S))}^{\cap}$ . If we assume that Q_s ⊈ Q_t, then C ∈ (SC (S) ∪ SO (S)) satisfies P_s ⊈ C ⊈ Q_t by (iii). If C ∈ SC (S) then C ⊈ Q_t ⇒ P_t ⊆ C ⇒ scl (P_t) ⊆ C, and we have scl (P_s) ⊈ scl (P_t) . If C ∈ SO (S) then P_s ⊈ C ⇒ C ⊆ Q_s ⇒ C ⊆ sint (Q_s), and so sint (Q_s) ⊈ sint (Q_t) . Hence (iii)⇒(iv).

Conversely, if (iv) satisfied and P_s ⊈ P_t then scl (P_s) ⊈ scl (P_t) or sint (Q_s) ⊈ sint (Q_t) . In the first case, P_t ⊆ scl (P_t) ⊆ Q_s and in the other case P_t ⊆ sint (Q_s) ⊆ Q_s. 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 $(S, S, τ, κ)$ be a ditopological space.

$(S, S, τ, κ)$ is semi-T₁-space iff it satisfies the conditions of Theorem 5.2 with $B = SC (S) .$ In particular, the following are characteristic of a semi-T₁ ditopological space.

For any $A \in S$ we have F_i ∈ SC (S) , i ∈ I satisfies A = ⋁ _i∈IF_i .

For all s, t ∈ S, if Q_s ⊈ Q_t then there exists F ∈ SC (S) satisfying P_s ⊈ F ⊈ Q_t .

$(S, S, τ, κ)$ is semi-co-T₁-space iff it satisfies the conditions of Theorem 5.1 with $A = SO (S) .$ In particular, the following are characteristic of a semi-co-T₁ ditopological space.

For any $A \in S$ we have G_i ∈ SO (S) , i ∈ I satisfies A = ⋂ _i∈IG_i .

For all s, t ∈ S, if Q_s ⊈ Q_t then there exists G ∈ SO (S) satisfying P_s ⊈ G ⊈ Q_t .

Proof. We prove (1), leaving the dual proof of (2) to the interested reader. Let $(S, S, τ, κ)$ be semi-T₀ and semi–R₀ . Take s, t ∈ S with Q_s ⊈ Q_t . According to Theorem 5.4 (iii), we get C ∈ (SC (S) ∪ SO (S)) satisfying P_s ⊈ C ⊈ Q_t . If C ∈ SC (S) then F = C ∈ SC (S) satisfies P_s ⊈ C ⊈ Q_t . Otherwise, if C ∈ SO (S) then C ⊈ Q_t implies scl (P_t) ⊆ C by semi-R₀, whence for F = scl (P_t) we have P_s ⊈ C ⊈ Q_t . This establishes Theorem 5.2 (ii) for $B = SC (S) .$ Conditions (a) and (b) are obtained from (i) and (ii) of Theorem 5.2, respectively.

Conversely, let $(S, S, τ, κ)$ satisfy Theorem 5.2 with $B = SC (S) .$ Then, it satisfies this theorem with $B = {(SC (S) \cup SO (S))}^{\cap}$ , so $(S, S, τ, κ)$ 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∈IF_i, F_i ∈ SC (S) , hence the result follows by [17, Theorem 4.2 (a) (2)]. □

Corollary 5.7. For a complemented ditopological space $(S, S, σ)$ the notions of semi-T₁ and semi-co-T₁coincide.

Proof. This is obvious by [17, Corollary 4.3]. Indeed, if A = ⋁ _j∈JF_j, F_j ∈ SC (S), then σ (A) = ⋂ _j∈Jσ (F_j) , σ (F_j) ∈ 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 $(S, S, τ, κ) .$

$(S, S, τ, κ)$ is semi-T₂ and semi-co-T₂ .

For all s, t ∈ S, if Q_s ⊈ Q_t then there exists H ∈ SO (S) and K ∈ SC (S) satisfying H ⊆ K, P_s ⊈ K and H ⊈ Q_t .

For $A \in S$ there exists $H_{i}^{j} \in SO (S),$ $K_{i}^{j} \in SC (S),$ j ∈ J, i ∈ I_j satisfying $H_{i}^{j} \subseteq K_{i}^{j}$ for all i, j and $A = ⋁_{j \in J} ⋂_{i \in I_{j}} H_{i}^{j} = ⋁_{j \in J} ⋂_{i \in I_{j}} K_{i}^{j} .$

Proof. (1) ⇒ (2) Assume that Q_s ⊈ Q_t. Since $(S, S, τ, κ)$ is semi-T₀ we have C ∈ (SC (S) ∪ SO (S)) with P_s ⊈ C ⊈ Q_t by Theorem 5.4 (iii). If C ∈ SO (S), then H ∈ SO (S) satisfies P_s ⊈ scl (H) and H ⊈ Q_t by the semi-R₁ axiom. Thus (2) is verified for K = scl (H). Otherwise, if C ∈ SC (S) then K ∈ SC (S) satisfies P_s ⊈ K, sint (K) ⊈ Q_t by the semi-co-R₁ axiom. Thus (2) is verified for sint (K) = H .

(2) ⇒ (3) We get A =⋂ { Q_s ∣ P_s ⊈ A } = ⋁ { P_t ∣ A ⊈ Q_t } for $A \in S .$ For s, t verifying A ⊈ Q_t and P_s ⊈ A we have Q_s ⊈ Q_t and so there exists $H_{t}^{s} \in SO (S),$ $K_{t}^{s} \in SC (S)$ with $H_{t}^{s} \subseteq K_{t}^{s}$ and $P_{s} ⊈ K_{t}^{s}$ , $H_{t}^{s} ⊈ Q_{t} .$ We obviously have $A = ⋁_{A ⊈ Q_{t}} ⋂_{P_{s} ⊈ A} H_{t}^{s} = ⋁_{A ⊈ Q_{t}} ⋂_{P_{s} ⊈ A} K_{t}^{s} .$

(3) ⇒ (1) Left to the reader.□

Proposition 5.10. For a complemented ditopological space $(S, S, σ),$ 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 $(S, S, τ, κ)$ is semi-coregular and semi-T₁ . Consider s, t ∈ S such that Q_s ⊈ Q_t. According to Theorem 5.6 (1)(b) we get F ∈ SC (S) such that P_s ⊈ F ⊈ Q_t, thus we have K ∈ SC (S) such that P_s ⊈ 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 ⊈ Q_t and H ⊆ K, thus $(S, S, τ, κ)$ is semi-T₂ and semi-co-T₂ by Theorem 5.9.□

Theorem 5.13. Let $(S, S, τ, κ)$ be the disjoint sum of non-empty disjoint ditopological texture spaces ( $S_{j}, S_{j}, τ_{j}, κ_{j}$ ), j ∈ J. Then $(S, S, τ, κ)$ is semi-T₀ iff ( $S_{j}, S_{j}, τ_{j}, κ_{j}$ ) is semi-T₀ whenever j ∈ J.

Proof. Let us assume that each ( $S_{j}, S_{j}, τ_{j}, κ_{j}$ ), j ∈ J is semi-T₀. Consider s, t ∈ S such that Q_s ⊈ Q_t . Let j, k ∈ J be the unique indices verifying s ∈ S_j, t ∈ S_k . If j ≠ k then $Q_{t}^{k} \neq S_{k}$ , thus C = S_k verifies C ∈ (SC (S) ∪ SO (S)) and P_s ⊈ C ⊈ Q_t . Otherwise, if j = k then $Q_{s}^{k} ⊈ Q_{t}^{k}$ and we have C_k ∈ (SC (S) ∪ SO (S)) _k with $P_{s}^{k} ⊈ C_{k} ⊈ Q_{t}^{k} .$ In this case, C = C_k satisfies C ∈ (SC (S) ∪ SO (S)) and P_s ⊈ C ⊈ Q_t . So $(S, S, τ, κ)$ is semi-T₀ by Theorem 5.4 (iii).

The converse is omitted.□

Corollary 5.14.Let $(S, S, τ, κ)$ be the disjoint sum of non-empty disjoint ditopological texture spaces ( $S_{j}, S_{j}, τ_{j}, κ_{j}$ ), j ∈ J. Then

$(S, S, τ, κ)$ is semi-T₁ (semi-co-T₁) iff (S_j, $S_{j},$ τ_j, κ_j) is semi-T₁ (semi-co-T₁) whenever j ∈ J.

$(S, S, τ, κ)$ is semi-T₂ (semi-co-T₂) iff (S_j, $S_{j},$ τ_j, κ_j) 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 $(S, S, τ, κ)$ be the product of non-empty disjoint ditopological texture spaces ( $S_{j}, S_{j}, τ_{j}$ , κ_j), j ∈ J . Then $(S, S, τ, κ)$ is semi-T₀ iff ( $S_{j}, S_{j}, τ_{j}$ , κ_j) is semi-T₀ whenever j ∈ J.

Proof. Let $(S, S, τ, κ)$ be semi-T₀. Consider any j ∈ J and for s_j, t_j ∈ S_j with Q_{s
_j} ⊈ Q_{t
_j} . Choose $u_{k} \in S_{k}^{♭}$ so that k ∈ J ∖ { j } , since S_k ≠ ∅ . Now let t = (t_i) and s = (s_i) ∈S defined by $s_{i} = {\begin{matrix} s_{j}, if i = j, \\ u_{i}, if i \neq j, \end{matrix} t_{i} = {\begin{matrix} t_{j}, if i = j, \\ u_{i}, if i \neq j . \end{matrix}$ It is imply that Q_s ⊈ Q_t, since Q_{s
_j} ⊈ Q_{t
_j} . By the Theorem 5.4 (iii), there exists C ∈ (SC (S) ∪ SO (S)) such that P_s ⊈ C ⊈ Q_t . Firstly, assume the case C ∈ SO (S) . According to the definition of the product topology, we get j₁, j₂, …, j_n ∈ J and C_{j
_k} ∈ SO (S_{j
_k}) , 1 ≤ k ≤ n, so that $⋃_{k = 1}^{n} E (j_{k}, C_{j_{k}}) \subseteq C$ and $⋃_{k = 1}^{n} E (j_{k}, C_{j_{k}}) ⊈ Q_{s}$ . Thus we get k, 1 ≤ k ≤ n, for which P_s ⊈ E (j_k, C_{j
_k}) ⊈ Q_t, so P_{s
_{j
_k}} ⊈ C_{j
_k} ⊈ Q_{t
_{j
_k}}. This leads to an contradiction if j_k ≠ j since then s_{j
_k} = t_{j
_k} . Then j_k = j, and we obtain C_j ∈ SO (S_j) satisfying P_{s
_j} ⊈ C_j ⊈ Q_{t
_j}.

The other case, for C ∈ SC (S), can be handled similarly above. Thus each ( $S_{j}, S_{j}, τ_{j}, κ_{j}$ ), j ∈ J is semi-T₀ by Theorem 5.4 (iii).

Conversely, let us assume that each ( $S_{j}, S_{j}, τ_{j}, κ_{j}$ ), j ∈ J is semi-T₀. Consider t = (t_j) and s = (s_j) ∈ S such that Q_s ⊈ Q_t. According to Theorem 5.4 (iii), it will be sufficient to find C ∈ (SC (S) ∪ SO (S)) such that P_s ⊈ C ⊈ Q_t. On the other hand because Q_s = ⋃ _j∈JE (j, Q_{s
_j}) ⊈ Q_t there exists j ∈ J such that E (j, Q_{s
_j}) ⊈ Q_t and so E (j, Q_{s
_j}) ⊈ E (j, Q_{t
_j}) , implying that Q_{s
_j} ⊈ Q_{t
_j} and by Theorem 5.4 (iii) we have C_j ∈ (SC (S) ∪ SO (S)) _j such that P_{s
_j} ⊈ C_j ⊈ Q_{t
_j} . Obviously C = E (j, C_j) has the required properties. Thus $(S, S, τ, κ)$ is semi-T₀ . □

Corollary 5.16.Let $(S, S, τ, κ)$ be the product of non-empty disjoint ditopological texture spaces ( $S_{j}, S_{j}, τ_{j}$ , κ_j), j ∈ J . Then

$(S, S, τ, κ)$ is semi-T₁ (semi-co-T₁) iff ( $S_{j}, S_{j}, τ_{j}$ , κ_j) is semi-T₁ (semi-co-T₁) whenever j ∈ J.

$(S, S, τ, κ)$ is semi-T₂ (semi-co-T₂) iff ( $S_{j}, S_{j}, τ_{j}$ , κ_j) is semi-T₂ (semi-co-T₂) whenever j ∈ J.

Proof. Clear by Theorems 5.15 and [17, Theorem 4.13].□

6 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 $semi - T_{3} \Rightarrow semi - T_{2} and semi - co - T_{3} \Rightarrow semi - co - T_{2} .$

According to [17, Corollary 9]kd, we have

Proposition 6.3. For a complemented ditopological space $(S, S, σ)$ 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 ⊈ Q_s, there exists a semi-bi-irresolute difunction $(f, F) : (S, S, τ, κ) \to (𝕀, I, τ_{𝕀}, κ_{𝕀})$ such that P_s ⊆ f^←P₀ and F^←Q₁ ⊆ G.

Completely semi-co-regular if given K ∈ SC (S) , P_s ⊈ K, there exists a semi-bi-irresolute difunction $(f, F) : (S, S, τ, κ) \to (𝕀, I, τ_{𝕀}, κ_{𝕀})$ such that K ⊆ f^←P₀ and F^←Q₁ ⊆ Q_s.

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- $T_{3 \frac{1}{2}}$ (respectively, semi-co- $T_{3 \frac{1}{2}}$ ).

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 $(S, S, τ, κ)$ is completely semi-regular. Take G ∈ SO (S) and s ∈ S such that G ⊈ Q_s . Let $(f, F) : (S, S, τ, κ) \to (𝕀, I, τ_{𝕀}, κ_{𝕀})$ be a semi-bi-irresolute difunction such that F^←Q₁ ⊆ G and P_s ⊆ f^←P₀. Let H = F^←Q_1/2. Hence H ∈ SO (S) because $Q_{1 / 2} \in SO (𝕀)$ and (f, F) is semi-irresolute. Obviously $P_{s} \subseteq f^{\leftarrow} P_{0} \subseteq f^{\leftarrow} Q_{1 / 2} \subseteq f^{\leftarrow} P_{1 / 2} \subseteq f^{\leftarrow} Q_{1} \subseteq G$ because f^←Q₁ = F^←Q₁. In fact f^←P_1/2∈ SC (S) because (f, F) is semi-co-irresolute and $P_{1 / 2} \in SC (𝕀),$ while f^←Q_1/2 = F^←Q_1/2 = H. Therefore we can obtain P_s ⊆ H ⊆ scl (H) ⊆ G . So $(S, S, τ, κ)$ 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- $T_{3 \frac{1}{2}} \Rightarrow$ semi-T₃ and semi-co- $T_{3 \frac{1}{2}} \Rightarrow$ semi-co-T₃.

Proposition 6.7. For a complemented ditopology $(S, S, σ, τ, κ)$ , the notions of complete semi-regularity and complete semi-co-regularity properties are equivalent.

Proof. Assume that the complemented ditopology $(S, S, σ, τ, κ)$ is completely semi-regular. Take K ∈ SC (S), s ∈ S, with P_s ⊈ K. By the definition of σ we have σ (K) ⊈ σ (P_s) . Choose t ∈ S satisfying σ (K) ⊈ Q_t and P_t ⊈ σ (P_s). By the complemented of $(S, S, σ, τ, κ),$ we have σ (K) ∈ SO (S) and there exists a semi-bi-irresolute difunction $(f, F) : (S, S, σ, τ, κ) \to (𝕀, I, ι, τ_{𝕀}, κ_{𝕀})$ such that P_t ⊆ f^←P₀ and F^←Q₁ ⊆ σ (K). If P_t ⊈ σ (P_s) verifies P_s ⊈ σ (P_t), then σ (f^←P₀) ⊆ σ (P_t) . Hence we have P_s ⊈ σ (f^←P₀) and therefore σ (f^←P₀) ⊆ Q_s. 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 $(f, F)^{'} = (F^{'}, f^{'}) : (S, S, σ, τ, κ) \to (𝕀, I, ι, τ_{𝕀}, κ_{𝕀})$ with K⊆ (F′) ^←P₀ and (f′) ^←Q₁ ⊆ Q_s. Clearly (f, F) ′ is semi-bi-irresolute, thus $(S, S, σ, τ, κ)$ 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 $(S, S)$ be a texture and $(T, T)$ be a texture for which $(T, \subseteq)$ is a chain and $(g_{k}, G_{k}) : (S, S) \to (T, T)$ difunctions for k = 1, 2, . . . , n. Hence there exists a difunction $(g, G) : (S, S) \to (T, T)$ for which $g^{\leftarrow} B = ⋂_{k = 1}^{n}$ $g_{k}^{\leftarrow} B$ , $G^{\leftarrow} B = ⋂_{k = 1}^{n}$ $G_{k}^{\leftarrow} B$ for all B∈ $T$ .

Proof. The mapping $θ : T \to S$ is defined by $θ (B) = ⋂_{k = 1}^{n}$ $g_{k}^{\leftarrow} B$ for B∈ $T$ . 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 (B_j) ⊆ θ (⋁ _j∈JB_j) for B_j∈ $T$ , j ∈ J . It remains to show that the reverse inclusion. For this, we have $\begin{matrix} θ (⋁_{j \in J} B_{j}) & = & ⋂_{k = 1}^{n} g_{k}^{\leftarrow} ⋁_{j \in J} B_{j} = \\ ⋂_{k = 1}^{n} ⋁_{j \in J} g_{k}^{\leftarrow} B_{j} & = & ⋁_{α \in J^{n}} ⋂_{k = 1}^{n} g_{k}^{\leftarrow} B_{α (k)} \end{matrix}$ by the preservation of joins by inverse images [5, Corollary 2.26] and $(S, \subseteq)$ is completely distributive. Consider α ∈ Jⁿ. Because $(T, \subseteq)$ is a chain there exists k with 1 ≤ k ≤ n such that the finite set {B_α(k)∣ k = 1, 2, . . . , n } contains a set B_{j
_α} satisfying B_α(k)⊆ B_{j
_α}. Hence $⋂_{k = 1}^{n} g_{k}^{\leftarrow} B_{α (k)} \subseteq ⋂_{k = 1}^{n} g_{k}^{\leftarrow} B_{j_{α}} = θ (B_{j_{α}}),$ so $\begin{matrix} θ (⋁_{j \in J} B_{j}) & = & ⋁_{α \in J^{n}} ⋂_{k = 1}^{n} g_{k}^{\leftarrow} B_{α (k)} \subseteq \\ ⋁_{α \in J^{n}} θ (B_{j_{α}}) & \subseteq & ⋁_{j \in J} θ (B_{j}), \end{matrix}$ as required.

The proof is completed by noting that the difunction $(g, G) = (f^{θ}, F^{θ}) : (S, S) \to (T, T)$ satisfying g^←B = θ (B) for all $B \in T$ , the remaining equality being a consequence of the equality of the inverse image and inverse co-image for difunctions.□

Theorem 6.9. Let ( $S_{j}, S_{j}, τ_{j}, κ_{j}$ ), j ∈ J, be ditopological texture spaces and $(f_{j}, F_{j}) : (S, S, τ, κ) \to$ ( $S_{j}, S_{j}$ ,τ_j, κ_j) difunctions. If the ditopologies ( $S_{j}, S_{j}$ , τ_j, κ_j) are completely semi-regular (completely semi-co-regular) for all j ∈ J, then the initial ditopology $(S, S, τ, κ)$ generated by the given spaces and difunctions is completely semi-regular (respectively, completely semi-co-regular).

Proof. To show that $(S, S, τ, κ)$ is completely semi-regular take H ∈ SO (S) and s ∈ S with H ⊈ Q_s. Hence there exists H_k ∈ S_k, k = 1, 2, . . . , n such that $⋂_{k = 1}^{n} F_{k}^{\leftarrow} H_{j} ⊈ Q_{s}$ and $⋂_{k = 1}^{n} F_{k}^{\leftarrow} H_{j} \subseteq H$ , therefore $F_{k}^{\leftarrow} H_{j} ⊈ Q_{s}$ for k = 1, 2, . . . , n and we get s_k ∈ S_k such that H_k ⊈ Q_{s
_k} and $P_{(s, s_{k})}^{_} ⊈ F_{k}$ . Since ( $S_{k}, S_{k}, τ_{k}, κ_{k}$ ) is completely semi-regular there exists a semi-bi-irresolute difunction (u_k, U_k) on $(S_{k}, S_{k}, τ_{k}, κ_{k})$ to $(𝕀, I, τ_{𝕀}, κ_{𝕀})$ for which $U_{k}^{\leftarrow} Q_{1} \subseteq H_{k}$ and $P_{s_{k}} \subseteq u_{k}^{\leftarrow} P_{0}$ . Let (g_k, G_k) = (u_k, U_k) ∘ (f_k, F_k). Then $(g_{k}, G_{k}) : (S, S, τ, κ) \to (𝕀, I, τ_{𝕀}, κ_{𝕀})$ for each k, and $(I, \subseteq)$ is also chain, thus by Lemma 6.8 there exists a difunction $(g, G) : (S, S, τ, κ) \to (𝕀, I, τ_{𝕀}, κ_{𝕀})$ satisfying $g^{\leftarrow} B = ⋂_{k = 1}^{n} g_{k}^{\leftarrow} B$ and $G^{\leftarrow} B = ⋂_{k = 1}^{n} G_{k}^{\leftarrow} B$ for all $B \in I$ . Obviously, for each k the semi-bi-irresolute of (f_k, F_k) and (u_k, U_k) imply that of (g_k, G_k). Thus (g, G) is semi-bi-irresolute.

It remains to show that G^←Q₁ ⊆ H and P_s ⊆ g^←P₀.

If now we suppose P_s⊈ g^←P₀ then for some k, 1 ≤ k ≤ n, we have P_s $⊈ g_{k}^{\leftarrow} P_{0} = f_{k}^{\leftarrow} u_{k}^{\leftarrow} P_{0}$ by [5, Lemma 2.16 (2)]bed1. Therefore we have $s_{k}^{'} \in S_{k}$ with f_k⊈ ${\bar{Q}}_{(s, s_{k}^{'})}$ and $P_{s_{k}^{'}} ⊈ u_{k}^{\leftarrow} P_{0}$ . From the definition of (f_k, F_k) we now have P_{s
_k}⊈ ${\bar{Q}}_{s_{k}^{'}}$ , so $P_{s_{k}^{'}} \subseteq P_{s_{k}}$ and we obtain the contradiction $P_{s_{k}} ⊈ u_{k}^{\leftarrow} P_{0}$ .

Thus $G^{\leftarrow} Q_{1} = ⋂_{k = 1}^{n} G_{k}^{\leftarrow} Q_{1} = ⋂_{k = 1}^{n} F_{k}^{\leftarrow} U_{k}^{\leftarrow} Q_{1} \subseteq ⋂_{k = 1}^{n} F_{k}^{\leftarrow} H_{k} \subseteq H$ . 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 $(S, S, τ, κ)$ be the product of non-empty disjoint ditopological texture spaces (S_j, $S_{j}$ ,τ_j, κ_j), j ∈ J . If ( $S_{j}, S_{j}, τ_{j}, κ_{j}$ ) is completely semi-regular (completely semi-co-regular), then $(S, S, τ, κ)$ is completely semi-regular (respectively, completely semi-co-regular).

Definition 6.11. A ditopological space $(S, S, τ, κ)$ 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 $(S, S, τ, κ)$ 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 $(S, S, τ, κ)$ . If SO (S) ⊆ SC (S) or SC (S) ⊆ SO (S) then $(S, S, τ, κ)$ is semi-normal. In particular, (SO (S), $S$ ) and (SC (S), $S$ ) are semi-normal.

The unit interval $(𝕀, I)$ 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 ⊈ Q_s. Hence scl (P_s) ⊆ G, and applying normality with F = scl (P_s) ∈ SC (S) gives H ∈ SO (S) with P_s ⊆ scl (P_s) ⊆ H ⊆ scl (H) ⊆ G. Thus $(S, S, τ, κ)$ 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 ( $S, S$ , τ, κ) is semi-normal iff given F ∈ SC (S), G ∈ SO (S) satisfying F ⊆ G there exists a semi-bi-irresolute difunction $(w, W) : (S, S, τ, κ) \to (𝕀, I, τ_{𝕀}, κ_{𝕀})$ such that F⊆ w^←P₀ and W^←Q₁ ⊆ G.

Proof. Assume that $(S, S, τ, κ)$ 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, $n \in ℕ^{+}$ , a set H (α) ∈ SO (S) satisfying $F \subseteq H (α) \subseteq scl (H (α^{'})) \subseteq G$ for all binary numbers α,α′ with 0 < α < α′ < 1. We now define a function $ω : S \to 𝕀$ by setting $ω (s) = inf {α ∣ H (α) ⊈ Q_{s}},$ where the infimum of an empty set of binary numbers is taken to be 1. If P_s ⊈ Q_s′ then P_s′ ⊆ P_s, therefore Q_s′ ⊆ Q_s and we deduce that ω (s′) ≤ ω (s). Thus P_ω(s) ⊈ Q_ω(s′), and we have shown that satisfies condition (a) of [5, Lemma 3.4]bed1. We deduce from this lemma that the equalities $\begin{matrix} w & = & ⋁ {{\bar{P}}_{(s, t)} ∣ \exists v \in S with P_{s} ⊈ Q_{v}, P_{ω (v)} ⊈ Q_{t}} \\ = & ⋁ {{\bar{P}}_{(s, ω (v))} ∣ v \in S, P_{s} ⊈ Q_{v}}, \end{matrix}$ (1) $\begin{matrix} W & = & ⋂ {{\bar{Q}}_{(s, t)} ∣ \exists v \in S with P_{v} ⊈ Q_{s}, P_{t} ⊈ Q_{ω (v)}} \\ = & ⋂ {{\bar{Q}}_{(s, ω (v))} ∣ v \in S, P_{v} ⊈ Q_{s}}, \end{matrix}$ (2) define a difunction (w, W) : $(S, S, τ, κ) \to (𝕀, I, τ_{𝕀}, κ_{𝕀})$ . For r∈ $𝕀$ it follows that $w^{\leftarrow} P_{r} = ⋂ {scl (H (α)) ∣ r < α},$ (3) $W^{\leftarrow} Q_{r} = ⋁ {H (α) ∣ α < r} .$ (4) We show that (3). Assume that w^←P_r⊈ ⋂{scl (H (α)) ∣ r < α} and take a binary number α with r < α and w^←P_r ⊈ scl (H (α)). Now, we have s ∈ S with w^←P_r ⊈ Q_s and P_s ⊈ scl (H (α)). From w^←P_r ⊈ Q_s, we have s′ ∈ S with P_s′ ⊈ Q_s for which $w ⊈ {\bar{Q}}_{(s^{'}, t)} \Rightarrow P_{t} \subseteq P_{r}$ for all $t \in 𝕀$ , whence we have ω (s) ≤ r because $w ⊈ {\bar{Q}}_{(s^{'}, ω (s))}$ by (1). Thus ω (s) < α and we take α′ with α′ < α and H (α′) ⊈ Q_s by the definition of ω. But this implies P_s ⊆ H (α′) ⊆ scl (H (α)), which is a contradiction.

If now we suppose ⋂ { scl (H (α)) ∣ r < α } ⊈ w^←P_r then we have s ∈ S satisfying ⋂{scl (H (α)) ∣ r < α ⊈ Q_s} and P_s ⊈ w^←P_r. Hence we have $t \in 𝕀$ with $w ⊈ {\bar{Q}}_{(s, t)}$ and P_t ⊈ P_r, that is r < t. From (1) we now have v ∈ S with P_s ⊈ Q_v and ${\bar{P}}_{(s, ω (v))} ⊈ {\bar{Q}}_{(s, t)}$ . 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′)) ⊈ Q_s, and we deduce from scl (H (α′)) ⊆ H (α) that H (α) ⊈ Q_s. 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 $SC (𝕀) ∖ {\emptyset}$ have the form P_r, $r \in 𝕀$ , and from (3) we have that w^←P_r ∈ 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 $SO (𝕀) ∖ {𝕀}$ have the form Q_r, $r \in 𝕀$ , and from (4) we have W^←Q_r ∈ SO (S). Thus (w, W) is semi-irresolute.

Conversely if $(w, W) : (S, S, τ, κ) \to (𝕀, I, τ_{𝕀}, κ_{𝕀})$ such that F⊆ w^←P₀ and W^←Q₁ ⊆ G then H = W^←Q_1/2 ∈ SO (S), K = w^←P_1/2 ∈ 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 $(S, S, τ, κ)$ be semi-R₀ and semi-normal. Take G ∈ SO (S), s ∈ S such that G ⊈ Q_s. Then scl (P_s) ⊆ G, and by Theorem 6.16 with F = scl (P_s) gives a semi-bi-irresolute difunction $(w, W) : (S, S, τ, κ) \to (𝕀, I, τ_{𝕀}, κ_{𝕀})$ satisfying W^←Q₁ ⊆ G and P_s ⊆ w^←P₀. Therefore $(S, S, τ, κ)$ is completely semi-regular. The other result can be shown dual to the above, and is omitted.□

Corollary 6.18.Semi-T₄⇒semi- $T_{3 \frac{1}{2}} \Rightarrow$ semi-T₃ and

semi-co-T₄⇒semi-co- $T_{3 \frac{1}{2}} \Rightarrow$ 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- $T_{3 \frac{1}{2}}$ . So semi-T₄⇒semi- $T_{3 \frac{1}{2}}$ , and semi- $T_{3 \frac{1}{2}} \Rightarrow$ semi-T₃ follows from Corollary 6.6. The remaining implications can be proved similarly.□

7 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 (μ_j, M_j) ∈ T, j = 1, 2, …, n then $⊓_{j = 1}^{n} (μ_{j}, M_{j})$ ∈T,

For any family of soft fuzzy sets if (μ_j, M_j) ∈ T, j ∈ J then ⊔_j∈J (μ_j, M_j) ∈ 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 $\bar{(μ, M)}$ and defined as follows: $\bar{(μ, M)} = ⊓ {(v, N) ∣ (μ, M) ⊑ (v, N) \in T^{'}} .$

Similarly the interior (μ, M) ° of (μ, M) is defined as follows: ${(μ, M)}^{\circ} = ⊔ {(v, N) ∣ (v, N) \in T, (v, N) ⊑ (μ, M)} .$

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 $(Z_{X}, Z_{X}, z_{X})$ . If T is a topology on X, then T ⊆ SF (X) and we may apply the isomorphism (μ, M) → A_ξ(μ,M) to give $A_{ξ (T)} \subseteq Z_{X}$ . Since A_ξ(0,∅) =∅, $A_{ξ (1, X)} = X \times 𝕀 = Z_{X}$ , and the isomorphism takes meet and join in SF (X) to intersection and union respectively in $Z_{X}$ , it is immediate that A_ξ(T) is a topology on the plain complemented texture $(Z_{X}, Z_{X}, z_{X})$ . On the other hand, for (μ, M) ∈ T we have A_{ξ((μ,M)′)} = z_X (A_ξ(μ,M)), so A_ξ(T′) = Z_X (A_ξ(T)) is a cotopology on $(Z_{X}, Z_{X}, z_{X})$ . Hence (A_ξ(T), A_ξ(T′)) is the complemented ditopology on $(Z_{X}, Z_{X}, z_{X})$ corresponding to T.

Remark 7.3. We observe that $A_{\bar{(μ, M)}}$ which is just the closure of A_ξ(μ,M) with respect to the ditopology (A_ξ(T), A_ξ(T′)). Similarly, A_ξ(μ,M)° is the interior of A_ξ(μ,M).

The category whose objects are complemented ditopological texture spaces of the form (Z_X, $Z_{X}$ ,z_X, τ_x, κ_x), 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 $G :$ 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.

7.1 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 $(μ, M) ⊑ \bar{{(μ, M)}^{\circ}}$ .

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 ${\bar{(μ, M)}}^{s}$ of the soft fuzzy set (μ, M) is defined as follows: $\begin{matrix} {\bar{(μ, M)}}^{s} & = & ⊓ {(v, N) ∣ (v, N) is SF - semi - closed, \\ (μ, M) & ⊑ & (v, N)} . \end{matrix}$ Likewise, the semi-interior (μ, M) ^∘_S of (μ, M) is defined as follows: $\begin{matrix} (μ, M)^{\circ_{S}} & = & ⊔ {(v, N) ∣ (v, N) is SF - semi - open, \\ (v, N) & ⊑ & (μ, M)} . \end{matrix}$ Obviously, (μ, M) ^∘_S is the greatest SF-semi-open set which is contained in (μ, M) and ${\bar{(μ, M)}}^{s}$ is the smallest SF-semi-closed set which contains (μ, M) and we have, $\begin{matrix} (μ, M) & ⊑ & {\bar{(μ, M)}}^{s} ⊑ \bar{(μ, M),} and \\ (μ, M)^{\circ} & ⊑ & (μ, M)^{\circ_{S}} ⊑ (μ, M) . \end{matrix}$

8 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 (A_ξ(T), A_ξ(T′)) on $(X \times 𝕀, P (X) \otimes I, σ_{X})$ is semi-R₀ iff (μ, M) SF-semi-open set, $(x_{s}, {x}) \in (μ, M) \Rightarrow {\bar{(x_{s}, {x})}}^{s} ⊑ (μ, M),$ and semi-co-R₀ iff (υ, N) SF-semi-closed set, $(x_{s}, {x}) \notin (υ, N) \Rightarrow (υ, N) ⊑ {(x^{s}, X ∖ {x})}^{\circ_{S}} .$

Proof. Under the isomorphism $G$ , (μ, M) SF-semi-open set corresponds to $A_{ξ (μ, M)} \in Z_{X}$ and (x_s, { x }) ∈ (μ, M) to P_(x,s) ⊆ A_ξ(μ,M), and then to A_ξ(μ,M) ⊈ Q_(x,s) because we deal with a plain texture. The semi-R₀ axiom gives scl (P_(x,s)) ⊆ A_ξ(μ,M), which corresponds to ${\bar{(x_{s}, {x})}}^{s} ⊑ (μ, M)$ , 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 (A_ξ(T), A_ξ(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 $(x_{s}, {x}) \in (μ, M) \Rightarrow {\bar{(x_{s}, {x})}}^{s} ⊑ (μ, M),$

(υ, N) SF-semi-closed set with (x_s, { x }) ∉ (υ, N) ⇒ (υ, N) ⊑ (x^s, 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 (υ_i, N_i) SF-semi-closed sets, i ∈ I, with (μ, M) = ⊔ _i∈I (υ_i, N_i) .

Given (μ, M) SF-semi-open set with (x_s, { x }) ∈ (μ, M) there exists, (υ, N) SF-semi-closed set with (υ, N) ⊑ (μ, M) and (x_s, { x }) ∈ (υ, N) .

For (υ, N) SF-semi-closed set there are (μ_i, M_i) SF-semi-open sets, i ∈ I, with (υ, N) = ⊓ _i∈I (μ_i, M_i) .

Given (υ, N) SF-semi-closed set with (x_s, { x }) ∉ (υ, N) there exists, (μ, M) SF-semi-open set with (μ, M) ⊑ (υ, N) and (x_s, { 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 (x_s, { x }) ∈ (μ, M) and (y_t, { y }) ∉ (μ, M) ⇒ ∃ (υ, N) SF-semi-open set with (x_s, { x }) ∈ (υ, N) and $(y_{t}, {y}) \notin {\bar{(υ, N)}}^{s} .$

(υ, N) SF-semi-closed set with (x_s, { x }) ∉ (υ, N) and (y_t, { y }) ∈ (υ, N) ⇒ ∃ (μ, M) SF-semi-closed set with (x_s, { x }) ∉ (μ, M) and (y_t, { 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 $(μ_{j}^{i}, M_{j}^{i})$ SF-semi-open sets, i ∈ I, j ∈ J_i, with $(μ, M) = ⊔_{i \in I} ⊓_{j \in J_{i}} (μ_{j}^{i}, M_{j}^{i}) = ⊔_{i \in I} ⊓_{j \in J_{i}} {\bar{(μ_{j}^{i}, M_{j}^{i})}}^{s} .$

Given (μ, M) SF-semi-open set with (x_s, { x }) ∈ (μ, M) and (y_t, { y }) ∉ (μ, M) , then we have (υ, N) SF-semi-open set with (x_s, { x }) ∈ (υ, N) $⊑ {\bar{(υ, N)}}^{s} ⊑ (y^{t}, X ∖ {y}) .$

For (υ, N) SF-semi-closed set we have $(υ_{j}^{i}, N_{j}^{i})$ SF-semi-closed sets, i ∈ I, j ∈ J_i, with $(υ, N) = ⊓_{i \in I} ⊔_{j \in J_{i}} (υ_{j}^{i}, N_{j}^{i}) = ⊓_{i \in I} ⊔_{j \in J_{i}} {(υ_{j}^{i}, N_{j}^{i})}^{\circ_{S}} .$

Given (υ, N) SF-semi-closed set with (x_s, { x }) ∉ (υ, N) and (y_t, { y }) ∈ (υ, N) , then we have (μ, M) SF-semi-closed set with (y_t, { y }) ∈ (μ, M) ^∘_S ⊑ (υ, N) ⊑ (x^s, 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 (x_s, { x }) ∈ (μ, M) ⇒ ∃ (υ, N) SF-semi-open set with (x_s, { x }) ∈ (υ, N) and ${\bar{(υ, N)}}^{s} ⊑ (μ, M) .$

(υ, N) SF-semi-closed set with (x_s, { x }) ∉ (υ, N) ⇒ ∃ (μ, M) SF-semi-closed set with (x_s, { 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 (μ_i, M_i) SF-semi-open sets, i ∈ I, with $(μ, M) = ⊔_{i \in I} (μ_{i}, M_{i}) = ⊔_{i \in I} {\bar{(μ_{i}, M_{i})}}^{s} .$

For (υ, N) SF-semi-closed set we have (υ_i, N_i) SF-semi-closed sets, i ∈ I, with (υ, N) = ⊓ _i∈I (υ_i, N_i) = ⊓ _i∈I (υ_i, N_i) ^∘_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:

(x_s, { x }) ∉ (y_t, { y }) ⇒ ∃ (μ_i, M_i)∈(SC (S) ∪ SO (S)), i ∈ I satisfying (y_t, { y }) ∈ ⊔ _i∈I(μ_i, M_i) ⊑ (x^s, X ∖ { x }) .

(x_s, { x }) ∉ (y_t, { y }) ⇒ ∃ (μ_i, M_i)∈(SC (S) ∪ SO (S)), i ∈ I satisfying (y_t, { y }) ∈ ⊓ _i∈I(μ_i, M_i) ⊑ (x^s, X ∖ { x }) .

(x_s, { x }) ∉ (y_t, { y }) ⇒ ∃ (μ, M)∈(SC (S) ∪ SO (S)) satisfying (x_s, { x }) ∉ (μ, M) and (y_t, { y }) ∈ (μ, M) .

${\bar{(x_{s}, {x})}}^{s} ⊑ {\bar{(y_{t}, {y})}}^{s}$ and (x^s, X ∖ { x }) ^∘_S ⊑ (y^t, X ∖ { y }) ^∘_S ⇒ (x^s, X ∖ { x }) ∈ (y^t, 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 (υ_i, N_i) SF-semi-closed sets, i ∈ I, with (υ, N) = ⊔ _i∈I (υ_i, N_i) .

(x_s, { x }) ∉ (y_t, { y }) ⇒ ∃ (υ, N) SF-semi-closed set satisfying (x_s, { x }) ∉ (υ, N) ∉ (y^t, X ∖ { y }) .

For any (μ, M) ∈ SF (X) we have (μ_i, M_i) SF-semi-open sets, i ∈ I, with (μ, M) = ⊓ _i∈I (μ_i, M_i) .

(x_s, { x }) ∉ (y_t, { y }) ⇒ ∃ (μ, M) SF-semi-open set satisfying (x_s, { x }) ∉ (μ, M) ∉ (y^t, 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:

(x_s, { x }) ∉ (y_t, { y }) ⇒ ∃ (μ, M) SF-semi-open set and ∃ (υ, N) SF-semi-closed set with (μ, M) ⊑ (υ, N) , (x_s, { x }) ∉ (υ, N) and (y_t, { y }) ∈ (μ, M) .

For (μ, M) ∈ SF (X) there exist $(μ_{j}^{i}, M_{j}^{i})$ SF-semi-open sets and $(υ_{j}^{i}, N_{j}^{i})$ SF-semi-closed sets, i ∈ I, j ∈ J_i, with $(μ_{j}^{i}, M_{j}^{i}) ⊑ (υ_{j}^{i}, N_{j}^{i})$ for all i, j so that $(μ, M) = ⊔_{i \in I} ⊓_{j \in J_{i}} (μ_{j}^{i}, M_{j}^{i}) = ⊔_{i \in I} ⊓_{j \in J_{i}} (υ_{j}^{i}, N_{j}^{i}) .$

Definition 8.13. A SF-topology T on X is soft fuzzy completely semi-regular if given (μ, M) SF-semi-open set with (x_s, { x }) ∈ (μ, M) there exists a inverse image $φ^{\leftarrow} : SF (𝕀) \to SF (X)$ satisfying (x_s, { x }) ⊑ φ^← (0_s, { 0 }) and $φ^{\leftarrow} (1^{t}, 𝕀 ∖ {1}) ⊑ (μ, M)$ .

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 $(υ, N) ⊑ (φ, G) ⊑ {\bar{(φ, G)}}^{s} ⊑ (μ, M) .$

9 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.

References

Aygünoğlu

and Aygün

, Some notes on soft topological spaces, Neural Comput and Applic21(1) (2012), 113–119.

Brown

L.M.

and Diker

, Ditopological texture spaces and intuitionistic sets, Fuzzy Sets and Systems98 (1998), 217–224.

Brown

L.M.

and Ertürk

, Fuzzy sets as texture spaces, I. Representation theorems, Fuzzy Sets and Systems110(2) (2000), 227–236.

Brown

L.M.

and Ertürk

, Fuzzy sets as texture spaces, II. Subtextures and quotient textures, Fuzzy Sets and Systems110(2) (2000), 237–245.

Brown

L.M.

, Ertürk

and Dost

Ş.

, Ditopological texture spaces and fuzzy topology, I. Basic concepts, Fuzzy Sets and Systems147(2) (2004), 171–199.

Brown

L.M.

, Ertürk

and Dost

Ş.

, Ditopological texture spaces and fuzzy topology, II. Topological considerations, Fuzzy Sets and Systems147(2) (2004), 201–231.

Brown

L.M.

, Ertürk

and Dost

Ş.

, Ditopological texture spaces and fuzzy topology, III. Separation axioms, Fuzzy Sets and Systems157(14) (2006), 1886–1912.

Brown

L.M.

and Gohar

, Compactness in ditopological texture spaces, Hacettepe J Math Stat38(1) (2009), 21–43.

Chen

, Soft semi-open and related properties in soft topological spaces, Appl Math Inf Sci7(1) (2013), 287–294.

10.

Dorsett

Ch.

, Semi compactness, semi separation axioms, and product spaces, Bull Malaysian Math Soc2(1) (1982), 11–17.

11.

Dorsett

Ch.

, semi-T2, semi-R1, and semi-R0 topological spaces, Ann Soc Sci Bruxelles92 (1978), 143–150.

12.

Dorsett

Ch.

, Semi-regular spaces, Soochow Journal of Mathematics8 (1982), 45–53.

13.

Dorsett

Ch.

, Semi-regular and semi-normal spaces, Soochow Journal of Mathematics15 (1989), 223–231.

14.

Dost

Ş.

, Ş. Dost, semi-open and semi-closed sets in ditopological texture spaces, J Adv Math Stud5(1) (2012), 97–110.

15.

Dost

Ş.

, Brown

L.M.

and Ertürk

, β -openandβ-closed sets in ditopological setting, Filomat24(2) (2010), 11–26.

16.

Dost

Ş.

, A Textural view of suft fuzzy rough sets, Journal of Intelligent and Fuzzy Systems28 (2015), 2519–2535.

17.

Kule

and Dost

Ş.

, Semi-regularity axioms in ditopological texture spaces, submitted.

18.

Levine

, Semi open sets and semi continuity in topological spaces, Amer Math Monthly70 (1963), 36–41.

19.

Maheshwari

S.N.

and Prasad

, Some new separation axioms, Ann Soc Sci Bruxelles89(1975), 395–402.

20.

Maji

P.K.

, Biswas

and Roy

A.R.

, Fuzzy soft sets, J Fuzzy Math9(3) (2001), 589–602.

21.

Molodtsov

D.A.

, Soft set theory-first results, Computer and Mathematics with Applications37 (1999), 19–31.

22.

Pei

and Miao

, From soft sets to information systems, Granular Computing, IEEE International Conference (2005), 617–621.

23.

Shabir

and Naz

, On soft topological spaces, Computer and Mathematics with Applications61 (2011), 1768–1799.

24.

Tiryaki

I.U.

, Fuzzy sets over the poset I, Hacettepe J Math Stat37(2) (2008), 143–166.

25.

Tiryaki

I.U.

and Brown

L.M.

, Plain textures and fuzzy sets via posets, Topology and its Applications158(15) (2011), 2005–2015.