Semi-compactness in ditopological texture spaces and soft fuzzy topological spaces

Abstract

The basic motivation of the theory of textures is to find a convenient point-set based setting for fuzzy sets. In some recent works on textures show that they also provide a fruitful model for soft fuzzy sets. Further the category of soft fuzzy topological texture spaces is isomorphic to the category of ditopological texture spaces. The first aim of this paper is to introduce and study the notions of semi-compactness and semi-stability in ditopological texture spaces. Secondly, it is given a natural link for these concepts in soft fuzzy topological spaces by using the mentioned isomorphism.

Keywords

Texture ditopology semi-compactness semi-stable soft fuzzy topology

1 Introduction

Soft set theory was introduced by Molodtsov [14] as a new approach for modeling uncertainties. The relationship between soft sets and information systems was discussed in [17]. Shabir and Naz [19] gave the concept of soft topological spaces and studied soft neighborhood of a point, soft separation axioms. On the other hand, initial soft topologies and soft compactness were given in [1]. In recent studies, the notions of semi-open sets with related properties [8] and soft semi-compactness [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 –4]. Ditopologies on textures unify the fuzzy topologies, topologies and bitopologies in a non-complemented setting by means of duality in the textural concepts [6, 7]. 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.

Now let X be a set and μ an $𝕀 = [0, 1]$ -fuzzy subset of X, M ⊆ X. Then the pair (μ, M) will be called a soft fuzzy subset of X. The set of all soft fuzzy subsets of X is denoted by SF (X). It was proved in [21] 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 $I = {[0, r), [0, r] ∣ r \in 𝕀}$ . 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.

The first aim of this study is to present a discussion on the concepts of semi-compactness and semi-stability in ditopological texture spaces 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.

2 Basic concepts

An overview on texture spaces and difunctions is given in the this section, and the reader is referred to [2 –7] for more background material.

Definition 2.1. Let S be a set. A texturing is a point-separating, complete, completely distributive lattice with respect to inclusion, which contains S and ∅, and for which arbitrary meets coincide with intersections, and finite joins with unions. If $S$ is a texturing of S, then the pair $(S, S)$ is called a texture space or shortly, texture.

For u ∈ $S$ , the p-sets and the q-sets are defined by

$\begin{matrix} P_{u} & = & ⋂ {A \in S ∣ u \in A}, \\ Q_{u} & = & ⋁ {A \in S ∣ u \notin A}, respectively . \end{matrix}$

In general, a texturing of S need not be closed under set complementation, but it may be that there exists a mapping $σ : S \to S$ satisfying σ (σ (A)) = A, $\forall A \in S$ and A ⊆ B ⇒ σ (B) ⊆ σ (A), $\forall A, B \in S$ . In this case σ is called a complementation on $(S, S)$ and $(S, S, σ)$ is said to be a complemented texture.

Example 2.2. For any set X, $(X, P (X), π)$ , π (Y) = X \ Y for Y ⊆ X, is the complemented discrete texture representing the usual set structure of X. Clearly, P_x = {x}, Q_x = X \ {x} for all x ∈ X.

(2) (See [3]) Let $(𝕃,')$ be a Hutton algebra, that is a complete, completely distributive lattice $𝕃$ equipped with an order-reversing involution ′. If we denote by $M_{𝕃}$ the set of molecules in $𝕃$ , set $\hat{a} = {m \in M_{𝕃} ∣ m \leq a}$ for $a \in 𝕃$ , and $M_{𝕃} = {\hat{a} ∣ a \in 𝕃}$ , then $(M_{𝕃}, M_{𝕃})$ is a texture. Moreover, $μ_{𝕃} : M_{𝕃} \to M_{𝕃}$ defined by $μ_{𝕃} (\hat{a}) = (\hat{a^{'}})$ is a complementation on $(M_{𝕃}, M_{𝕃})$ . We will refer to the complemented texture $(M_{𝕃}, M_{𝕃}, μ_{𝕃})$ as the Hutton texture of $(𝕃,')$ .

(3) Let L = (0, 1], $L = {(0, r] ∣ r \in [0, 1]}$ and λ ((0, r]) = (0, 1 - r], r ∈ [0, 1]. Clearly $(L, L, λ)$ is the Hutton texture of $(𝕀,')$ , where $𝕀 = [0, 1]$ with its usual order and r′ = 1 - r for $r \in 𝕀$ . Here P_r = Q_r = (0, r] for all r ∈ L.

(4) (See [3]) Let X be a non-empty set and $(𝕃,')$ a fixed Hutton algebra. Then the molecules of the Hutton algebra $𝕎_{X} = 𝕃^{X}$ of $𝕃$ -valued subsets of X are just the “fuzzy points” x_m for x ∈ X and $m \in M_{𝕃}$ . These are in one to one correspondence with the points of $W_{X} = X \times M_{𝕃}$ , and for $f \in 𝕎_{X}$ we now have $\hat{f} = {(x, m) \in W_{X} ∣ m \leq f (x)}$ . Setting $W_{X} = {\hat{f} ∣ f \in 𝕎_{X}}$ and $ω_{X} (\hat{f}) = \hat{(f^{'})}$ , f′ (x) = f (x) ′,gives us the Hutton texture $(W_{X}, W_{X}, ω_{X})$ of $𝕃^{X}$ .

(5) 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]. $(𝕀, I, ι)$ is a complemented texture, which we will refer to as the unit interval texture. Here P_t = [0, t] and Q_t = [0, t) for all $t \in 𝕀$ .

Definition 2.3. A ditopology on a texture $(S, S)$ is a pair (τ, κ) of subsets of $S$ , where the set of open sets τ and the set of closed sets κ satisfies $\begin{array}{l} S, \emptyset \in τ, S, \emptyset \in κ \\ G_{1}, G_{2} \in τ \Rightarrow G_{1} \cap G_{2} \in τ, K_{1}, K_{2} \in κ \Rightarrow K_{1} \cup K_{2} \in κ \\ G_{i} \in τ, i \in I \Rightarrow \underset{i \in I}{\lor} G_{i} \in τ, K_{i} \in κ, i \in I \Rightarrow \underset{i \in I}{\cap} K_{i} \in κ . \end{array}$

For $A \in S$ we define the closure and the interior 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)).

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 (for products see [3]). Clearly, ${\bar{P}}_{(s, t)} = {s} \times P_{t}$ and ${\bar{Q}}_{(s, t)} = (S \ {s} \times T) \cup (S \times Q_{t})$ where s ∈ S and t ∈ T.

Difunctions arise often in the study of textures and ditopological texture spaces. A difunction is a direlation [5] (f, F) satisfying certain additional conditions.

Definition 2.4. Let (f, F) be a direlation from $(S, S)$ to $(T, T)$ . Then (f, F) is called a difunction from $(S, S)$ to $(T, T)$ if it satisfies the following two conditions. $\begin{array}{l} D F 1 F o r s, s^{'} \in S, P_{s} \subseteq Q_{s^{'}} \Rightarrow \exists t \in T w i t h \\ f \subseteq {\bar{Q}}_{(s, t)} a n d {\bar{P}}_{(s', t)} \subseteq F \\ D F 2 F o r t, t^{'} \in T a n d s \in S, f \subseteq {\bar{Q}}_{(s, t)} a n d \\ {\bar{P}}_{(s, t')} \subseteq F \Rightarrow P_{t^{'}} \subseteq Q_{t} . \end{array}$

Definition 2.5. Let $(f, F) : (S, S) \to (T, T)$ be a difunction.

For $A \in S$ , the image f^→ (A) and the co-image F^→ (A) are defined by $\begin{array}{l} f^{\to} A = \cap {Q_{t} ∣ \forall s, f \subseteq {\bar{Q}}_{(s, t)} \Rightarrow A \subseteq Q_{s}}, \\ F^{\to} A = \lor {P_{t} ∣ \forall s, {\bar{P}}_{(s, t)} \subseteq F \Rightarrow P_{s} \subseteq A} . \end{array}$

For $B \in T$ , the inverse image f^← (B) and the inverse co-image F^← (B) are defined by $\begin{array}{l} f^{\leftarrow} B = \lor {P_{s} ∣ \forall t, f \subseteq {\bar{Q}}_{(s, t)} \Rightarrow P_{t} \subseteq B}, \\ F^{\leftarrow} B = \cap {Q_{s} ∣ \forall t, {\bar{P}}_{(s, t)} \subseteq F \Rightarrow B \subseteq Q_{t}} . \end{array}$

For a given difunction, the inverse image and the inverse co-image are equal; and the image and co-image are usually not.

Definition 2.6. Let $(f, F) : (S, S) \to (T, T)$ be a difunction. Then (f, F) is called surjective if it satisfies the condition

SUR. For $t, t^{'} \in T$ , $P_{t} \subseteq Q_{t^{'}} \Rightarrow \exists s \in S$ with $f \subseteq {\bar{Q}}_{(s, t')}$ and ${\bar{P}}_{(s, t)} \subseteq F$ .

Likewise, (f, F) is called injective if it satisfies the condition

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

If (f, F) is both injective and surjective then it is called bijective.

Note 2.7. In general, difunctions are not directly related to ordinary (point) functions between the base sets. We note that [5, Lemma 3.4]bed1 if $(S, S)$ , $(T, T)$ are textures and φ : S → T an ω-compatible point function, namely one satisfiying P_s ⊆ Q_s′ ⇒ P_φ(s) ⊆ Q_φ(s′), then the equalities $\begin{array}{l} f_{φ} = \lor {{\bar{P}}_{(s, t)} ∣ \exists u \in S satisfying \\ P_{s} \subseteq Q_{u} and P_{φ (u)} \subseteq Q_{t}}, \\ F_{φ} = \cap {{\bar{Q}}_{(s, t)} ∣ \exists u \in S satisfying \\ P_{u} \subseteq Q_{s} and P_{t} \subseteq Q_{φ (u)}}, \end{array}$ define a difunction $(f_{\overset{ˇ}{,}} F_{\overset{ˇ}{)}}$ on $(S, S)$ to $(T, T)$ . Moreover, for $B \in T$ ,, where φ^←B = ⋁ {P_s ∣ P_φ(u) ⊆ B ∀ u ∈ S with P_s ⊆ Q_u}.

We denote by ifTex the construct of textures and ω-preserving functions between the base sets.

The notions of semi-open sets, as a generalized of open sets, and semi-bicontinuity in ditopological texture spaces are introduced in [11].

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

semi-open if A ⊆ cl int (A),

semi-closed if int cl (A) ⊆ A.

We denote by SO (S) (SC (S)) the family of semi-open (semi-closed) sets in $(S, S, τ, κ)$ .

As usual, the semi-closure and the semi-interior of $A \in S$ under (τ, κ) are defined by the equalities $\begin{matrix} scl (A) & = & ⋂ {K \in SC (S) ∣ A \subseteq K}, \\ sint (A) & = & ⋁ {G \in SO (S) ∣ G \subseteq A} . \end{matrix}$

Definition 2.9. A difunction $(f, F) : (S, S, τ_{S}, κ_{S}) \to (T, T, τ_{T}, κ_{T})$ is called

semi-continuous (semi-irresolute) if B ∈ τ_T (B ∈ SO (T)) ⇒ F^← (B) ∈ SO (S),

semi-cocontinuous (semi-co-irresolute) if B ∈ κ_T (B ∈ SC (T)) ⇒ f^← (B) ∈ SC (S),

semi-bicontinuous (semi-bi-irresolute) if it is semi-continuous and semi-cocontinuous (semi-irresolute and semi-co-irresolute).

3 Semi-compactness in ditopological spaces

We observe that a ditopology is a “topology” for which there is no a priori relation between the open and closed sets. Hence some properties of topological spaces cannot take place in the ditopological texture space. As a result of this, dual properties occur in the ditopological texture space. For example, the ditopology (τ, κ) on $(S, S)$ is called compact (co-compact) [2] if every open cover (closed cocover) of S has a finite sub-cover (sub-cocover).

Here we recall from [2] that $C = {A_{j} \in S ∣ j \in J}$ is a cover of A (a cocover of A) if A ⊆ ⋁ A_j (⋂ _j∈JA_j ⊆ A).

In this section, we will deal with the notion of semi-compactness for the ditopological texture spaces. Firstly we recall that [10 , 20] a subset A of a topological space X is called semi-compact, if every cover {G_i ∣ i ∈ I} of A by semi-open sets G_i of X has a finite sub-cover. Now we give an analogous definition of semi-compactness in ditopological texture spaces. As expected, there is also the dual notion of semi-cocompactness.

Definition 3.1. Let (τ, κ) be a ditopology on the texture $(S, S)$ and $A \in S$ .

A is called semi-compact if every cover of A by semi-open sets has a finite sub-cover. In particular the ditopological texture space $(S, S, τ, κ)$ is called semi-compact if S is semi-compact.

A is called semi-cocompact if every cocover of A by semi-closed sets has a finite sub-cocover. In particular the ditopological texture space $(S, S, τ, κ)$ is called semi-cocompact if ∅ is semi-cocompact.

As can be seen in the following examples, semi-compactness and semi-cocompactness are independent.

Example 3.2. (1) We define a ditopology (τ, κ) on $(L, L)$ of Examples 2.2(3) where τ = {∅ , L} and $κ = L$ . Since the only semi-open sets are ∅ and L we see that (τ, κ) is semi-compact. However, it is not semi- cocompact, for example, the family $C = {(0, 1 / n] ∣ n = 1, 2, \dots}$ of semi-closed sets satisfies $⋂ C = \emptyset$ , but no finite subset of $C$ has an empty intersection.

(2) Dually, let $τ = L$ and κ = {∅ , L}. Then the ditopology (τ, κ) is semi-cocompact but not semi-compact.

(3) Let $(X, T)$ be a topological space. Then $(T, T^{c})$ is a ditopology on the complemented discrete texture space $(X, P (X), π_{X})$ , where $T^{c} = {π_{X} (G) = X \ G ∣ G \in T}$ . Clearly if $(X, T)$ is semi-compact then $(X, P (X), π_{X}, T, T^{c})$ is semi-compact and semi-cocompact.

The concepts of semi-compactness and semi-cocompactness are equivalent for complemented ditopological texture spaces.

Proposition 3.3. Let (τ, κ) be a complemented ditopology on $(S, S, σ)$ . Then $(S, S, σ, τ, κ)$ is semi-compact if and only if it is semi-cocompact.

Proof. Let (τ, κ) be a semi-cocompact ditopology. Consider the family of semi-open sets $C = {G_{j} ∣ j \in J}$ with $⋁ C = S$ . Since (τ, κ) is complemented, $D = {σ (G_{j}) ∣ j \in J}$ is a family of semi-closed sets. We observe that $\begin{matrix} ⋂ D & = & ⋂ {σ (G_{j}) ∣ j \in J} \\ = & σ (⋁ {G_{j} ∣ j \in J}) = σ (S) = \emptyset, \end{matrix}$ and so we have J′ ⊆ J finite with ⋂ {σ (G_j)∣ j ∈ J′} = ∅. Hence ⋁ {G_j ∣ j ∈ J′} = S, and we see that (τ, κ) is semi-compact.

Using dual arguments, it can be obtain the other direction. □

The next example show that the reverse implication of “compact (cocompact)⇒ semi-compact (semi-cocompact)” can not be true in general.

Example 3.4. (τ, κ) be a ditopology on $(L, L)$ of Examples 2.2 (3) where τ = κ = {L, ∅}. Clearly, every set in $L$ is semi-open and $C = {(0, 1 - 1 / n] ∣ n = 2, 3, \dots}$ is a cover of L by semi-open. Then $C$ has no finite sub-cover, so (τ, κ) is not semi-compact. However, (τ, κ) is compact since τ is finite.

Dually, (τ, κ) is cocompact but not semi-cocompact.

Note that the strong compactness (strong cocompactness) [13] is defined by the same way using pre-open (pre-closed) sets. But, since there is no relation [11] between semi-open(-closed) sets and pre-open(-closed) sets we have no any relation between semi-(co)-compactness and strong-(co)-compactness.

Theorem 3.5. Let $(f, F) : (S_{1}, S_{1}, τ_{1}, κ_{1}) \to (S_{2}, S_{2}, τ_{2}, κ_{2})$ be a semi-irresolute difunction. If $A \in S_{1}$ is semi-compact then $f^{\to} A \in S_{2}$ is semi-compact.

Proof. Let $C = {G_{j} ∣ j \in J}$ be a family of semi-open sets in $S_{2}$ and $f^{\to} (A) \subseteq \lor C = \underset{j \in J}{\lor} G_{j}$ Since F^← preserves arbitrary joins and A ⊆ F^← (f^→ (A)) by [5, Theorem 2.24 (2 a) and Corollary 2.12 (2)]bed1, we may write $A \subseteq F^{\leftarrow} (f^{\to} (A)) \subseteq F^{\leftarrow} (⋁_{j \in J} G_{j}) = ⋁_{j \in J} F^{\leftarrow} (G_{j}) .$

By semi-irresolutness, F^← (G_j) ∈ SO (S₁), j ∈ J and so by the semi-compactness of A there exists J′ ⊆ J finite such that Ase ⋃ _j∈J′F^← (G_j). Hence, by [5, Corollary 2.12 (2) and Theorem 2.24 (2 b)]bed1 $f^{\to} (A) \subseteq f^{\to} (\underset{j \in J^{'}}{\cup} F^{\leftarrow} (G_{j})) = \underset{j \in J^{'}}{\cup} f^{\to} (F^{\leftarrow} (G_{j})) \subseteq \underset{j \in J^{'}}{\cup} G_{j} .$

Thus f^→A is semi-compact. □

Proposition 3.6. Let $(f, F) : (S_{1}, S_{1}, τ_{1}, κ_{1}) \to (S_{2}, S_{2}, τ_{2}, κ_{2})$ be a semi-irresolute surjective difunction. If $(S_{1}, S_{1}, τ_{1}, κ_{1})$ is semi-compact then $(S_{2}, S_{2}, τ_{2}, κ_{2})$ is semi-compact.

Proof. Take A = S₁ in Theorem 3.5. By [5, Proposition 2.28 (1 c) and Corollary 2.33 (1)]bed1, we have f^→S₁ = f^→ (F^←S₂) = S₂ and so the proof is obtained immediately. □

As expected, we have dual results for semi-cocompactness. We omit the proofs.

Proposition 3.7. Let $(f, F) : (S_{1}, S_{1}, τ_{1}, κ_{1}) \to (S_{2}, S_{2}, τ_{2}, κ_{2})$ be a semi-co-irresolute difunction.

If $A \in S_{1}$ is semi- cocompact then $F^{\to} A \in S_{2}$ is semi-cocompact.

If (f, F) is surjective and $(S_{1}, S_{1}, τ_{1}, κ_{1})$ is semi-cocompact then $(S_{2}, S_{2}, τ_{2}, κ_{2})$ is semi-cocompact.

Let $(S_{j}, S_{j}, τ_{j}, κ_{j})_{j \in J}$ 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 $(S, S)$ to $(S_{j}, S_{j})$ [5, Lemma 3.9]bed1. On the other hand the ditopology (τ, κ) on $(S, S)$ with subbase ${Π_{j}^{\leftarrow} G = E (j, G) ∣ G \in τ_{j}, j \in J}$ and cosubbase ${π_{j}^{\leftarrow} K = E (j, K) ∣ K \in κ_{j}, j \in J}$ 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.

By [12, Lemma 2.11]go every projection difunction is surjective. Because of [6, Proposition 3.21]bed2 and [11, Proposition 2.13]ds1, the projection difunction is semi-bi-irresolute. As a result, from Propositions 3.6 and 3.7, we have immediately:

Corollary 3.8. Let $(S_{j}, S_{j}, τ_{j}, κ_{j})_{j \in J}$ be ditopological texture spaces and $(S, S, τ, κ)$ their product.

If $(S, S, τ, κ)$ is semi-compact then $(S_{j}, S_{j}, τ_{j}, κ_{j})$ is semi-compact for each j ∈ J.

If $(S, S, τ, κ)$ is semi-cocompact then $(S_{j}, S_{j}, τ_{j}, κ_{j})$ is semi-cocompact for each j ∈ J.

4 Semi-stability in ditopological spaces

Semi-compact ditopological texture spaces lack many of the some properties of semi-compact topological spaces. For example, semi-closed (semi-open) sets need not be semi-compact (semi-cocompact) in semi-compact (semi-cocompact) ditopological texture spaces. This leads to the following concepts.

Definition 4.1. Let $(S, S, τ, κ)$ be a ditopology. Then (τ, κ) is called

semi-stable if every semi-closed set $F \in S \ {S}$ is semi-compact in S.

semi-costable if every semi-open set $G \in S \ {\emptyset}$ is semi-cocompact in S.

As we will see in the next examples, semi-stability (semi-costability) are unrelated to semi-compactness (semi-cocompactness), respectively.

Example 4.2. Consider the texture $(L, L)$ of Examples 2.2 (3).

(1) Take $τ = L$ and κ = {∅ , L}. So, the ditopology (τ, κ) is not semi-compact because it is not compact. On the other hand (τ, κ) is semi-stable because in this space every semi-closed set is closed and the only closed set different from L is ∅, which is trivially semi-compact. Dually, if we take τ = {∅ , L} and $κ = L$ then this ditopology is semi-costable but not semi-cocompact.

(2) Take a ditopology (τ, κ) on $(L, L)$ where τ = {(0, s] ∣0 ≤ s ≤ 1/2} ∪ {L} and κ = {(0, s] ∣1/2 ≤ s ≤ 1} ∪ {∅}. If we take s ∈ L with 1/2 < s < 1 and set A = (0, s], then cl (A) = A and so int cl (A) = int (A) = (0, 1/2] ∈ κ, from which we see A ⊆ cl int (A), that is A is not semi-open. It follows that the only semi-open sets are the open sets, so (τ, κ) is semi-compact because any open cover of L must contain L. On the other hand the set (0, 1/2] is closed, and hence semi-closed, and it is clearly not compact so not semi-compact. It follows that (τ, κ) is not semi-stable. Using dual arguments, it can be show that (τ, κ) is also semi-cocompact but not semi-costable.

Clearly, semi-stability is generalized of stability [12] in ditopological texture spaces. The strong (co)-stability [13] is given by using pre-open (-closed) sets. As expected, there is no relation between semi-(co)-stability and strong-(co)-stability. The following examples establish that the some implications also cannot be reversed in general.

Example 4.3. Consider the unit interval texture $(𝕀, I)$ of Examples 2.2 (5), let ι be the complementation ι ([0, r)) = [0, 1 - r], ι ([0, r]) = [0, 1 - r), and $(τ_{𝕀}, κ_{𝕀})$ the standard complemented ditopology given by $\begin{matrix} τ_{𝕀} & = & {[0, r) ∣ r \in 𝕀} \cup {𝕀}, \\ κ_{𝕀} & = & {[0, r] ∣ r \in 𝕀} \cup {\emptyset} . \end{matrix}$

The ditopology $(τ_{𝕀}, κ_{𝕀})$ is strongly stable and strongly costable since the family of pre-open sets is $τ_{𝕀}$ and the family of pre-closed sets is $κ_{𝕀}$ . Hence it is stable and costable by [12]. On the other hand, $SO (𝕀) = SC (𝕀) = I$ and this space is neither semi-compact nor semi-cocompact. For example, A = [0, 1) is a semi-closed set and the family $C = {[0, 1 - 1 / n) ∣ n = 1, 2, \dots}$ of semi-open sets is a cover of A, but no finite sub-cover of $C$ . Similarly, it can be show that this space is not semi-costable.

The below can be seen that independent of each the concepts of semi-stability and semi-costability.

Example 4.4. Consider the texture $(L, L)$ .

We take τ = {∅ , (0, 1/3] , L} and $κ = L$ . In this space, semi-open sets are only the open sets. Since τ is finite every semi-closed set is, automatically, semi-compact. Hence (τ, κ) is semi-stable and also semi-compact. However the semi-open set (0, 1/3] is not semi-cocompact since $C = {(0, 1 / 3 + 1 / n] ∣ n = 4, 5, \dots}$ is a semi-closed cocover with no finite sub-cocover. Hence (τ, κ) is not semi-costable. It is also not semi-cocompact.

Dually, let $τ = L$ and κ = {∅ , (0, 1/3] , L}. The ditopology (τ, κ) is semi-costable and semi-cocompact but not semi-stable or semi-compact.

As expected, these concepts are equivalent for complemented ditopological texture spaces

Proposition 4.5. Let $(S, S, σ)$ be a complemented texture and let (τ, κ) be a complemented ditopology on $(S, S, σ)$ . Then (τ, κ) is semi-stable if and only if it is semi-costable.

Proof. Suppose that (τ, κ) be semi-costable. Take K ∈ SC (S) with K ≠ S. Let $C$ be a semi-open cover of K. Then σ (K) is semi-open and σ (K)≠ ∅. Hence σ (K) is semi-cocompact by assumption. Let $C^{*} = {σ (G) ∣ G \in C}$ . Since $K \subseteq ⋁ C$ we have $⋂ C^{*} \subseteq σ (K)$ , and so $C^{*}$ is a semi-closed cocover of σ (K). Hence there exists $G_{1}, G_{2}, \dots, G_{n} \in C$ so that $\begin{matrix} σ (G_{1}) \cap σ (G_{2}) \cap \dots \cap σ (G_{n}) \\ = σ (G_{1} \cup G_{2} \cup \dots \cup G_{n}) \subseteq σ (K) \end{matrix}$

This gives σ (σ (K)) = K ⊆ G₁ ∪ G₂ ∪ … ∪ G_n, so K is semi-compact in S. Hence (τ, κ) is semi-stable.

The proof that semi-stable implies semi-costable is obtained by using dual arguments. □

Theorem 4.6. Let (f, F) be a semi-bi-irresolute surjective difunction from $(S_{1}, S_{1}, τ_{1}, κ_{1})$ to $(S_{2}, S_{2}, τ_{2}, κ_{2})$ .

If (τ₁, κ₁) is semi-stable then (τ₂, κ₂) is semi-stable.

If (τ₁, κ₁) is semi-costable then (τ₂, κ₂) is semi-costable.

Proof. We prove (1), leaving the dual proof of (2) to the interested reader. Let K ∈ SC (S₂) with K ≠ S₂. Then f^←K ∈ SC (S₁) by semi-co-irresolutness. Now we show that f^←K ≠ S₁. Assume the contrary. Since f^←S₂ = S₁, by [5, Lemma 2.28 (1 c)]bed1 we have f^←S₂ ⊆ f^←K, whence S₂ ⊆ K by [5, Corollary 2.33 (1 ii)]bed1 as (f, F) is surjective. This is a contradiction, so f^← (K) ≠ S₁. Hence f^← (K) is semi-compact in $(S_{1}, S_{1}, τ_{1}, κ_{1})$ by semi-stability. As (f, F) is semi-irresolute, f^→ (f^←K) is semi-compact for the ditopology (τ₂, κ₂) by Theorem 3.5, and by [5, Corollary 2.33 (1)]bed1 this set is equal to K. This establishes that $(S_{2}, S_{2}, τ_{2}, κ_{2})$ is semi-stable. □

Since the projection difunction semi-bi-irresolute and surjective we give immediately the following results.

Corollary 4.7. Let $(S_{j}, S_{j}, τ_{j}, κ_{j})_{j \in J}$ be ditopological texture spaces and $(S, S, τ, κ)$ their product.

If $(S, S, τ, κ)$ is semi-stable then $(S_{j}, S_{j}, τ_{j}, κ_{j})$ is semi-stable for each j ∈ J.

If $(S, S, τ, κ)$ is semi-costable then $(S_{j}, S_{j}, τ_{j}, κ_{j})$ is semi-costable for each j ∈ J.

5 Semi-dicompactness in ditopological spaces

In the fourth section, the concepts of semi-compact and semi-cocompact ditopological texture spaces are discussed. Moreover semi-stable and semi-costable ditopological texture spaces have been introduced based on these concepts in the fifth section. Ditopological texture spaces which have all of the these four properties are discussed in this section under the name semi-dicompact.

Definition 5.1. A ditopological texture space is called semi-dicompact if it is semi-compact, semi-cocompact, semi-stable and semi-costable.

Theorem 5.2. Semi-dicompactness is preserved under a semi-bi-irresolute surjective difunction.

Proof. As a consequence of Propositions 3.6, 3.7 and Theorems 4.6, it is immediately obtained. □

To give non-trivial characterizations of semi-dicompactness, we need to recall some definitions from [2]: A difamily $D = {(G_{j}, F_{j}) ∣ j \in J}$ is called a dicover of $(S, S)$ if for all partitions J₁, J₂ of J (including the trivial partitions) we have $⋂_{j \in J_{1}} F_{j} \subseteq ⋁_{j \in J_{2}} G_{j} .$

A difamily $D$ has the finite exclusion property (fep) if whenever $(F_{j}, G_{j}) \in D$ , j = 1, 2, …, n we have $⋂_{j = 1}^{n} F_{j} ⊈ ⋃_{j = 1}^{n} G_{j}$ .

Now, we adopt the following concepts from [2].

Definition 5.3. Let (τ, κ) be a ditopology on $(S, S)$ .

A set $D \subseteq S \times S$ is called a difamily on $(S, S)$ . A difamily $D$ satisfying $D \subseteq SO (S) \times SC (S)$ is semi-open, semi-co-closed, one satisfying $D \subseteq SC (S) \times SO (S)$ is semi-closed, semi-co-open.

A semi-closed, semi-co-open difamily $D$ with $⋂ {F ∣ F \in dom D} ⊈ ⋁ {G ∣ G \in ran D}$ is said to be bound in $(S, S, τ, κ)$ .

Theorem 5.4. For a ditopological texture space $(S, S, τ, κ)$ the following are equivalent:

$(S, S, τ, κ)$ is semi-dicompact.

Every semi-closed, semi-co-open difamily with the finite exclusion property is bound.

Every semi-open, semi-co-closed dicover has a finite sub-dicover.

Proof. (1)implies(2) Let (τ, κ) be a semi-dicompact ditopology on $(S, S)$ . Now suppose that we have a semi-closed, semi-co-open difamily $B = {(F_{j}, G_{j}) ∣ j \in J}$ with the fep, which is not bound in $(S, S, τ, κ)$ . Let F = ⋂ _i∈IF_i. Then F is semi-closed by [11, Lemma 2.3 (iii)]ds1 and Fse ⋁ _i∈IG_i since $B$ is not bound. According as F ≠ S or F = S we may use semi-stability or semi-compactness, respectively, to show the existence of a finite subset J₁ of J with Fse ⋃ _j∈J₁G_j. Now let G = ⋃ _j∈J₁G_j. By [11, Lemma 2.3 (ii)]ds1, G is a semi-open set. Also, ⋂_j∈JF_j ⊆ G. Hence, according as G≠ ∅ or G =∅, we may use semi-costability or semi-cocompactness, respectively, to show that ⋂_j∈J₂F_j ⊆ G for some finite subset J₂ of J. Since now ⋂_{j∈J₁∪J₂}F_j ⊆ ⋃ _{j∈J₁∪J₂}G_j we have a contradiction to the fact that $B$ has the fep.

(2)implies(3) Suppose that $C = {(G_{i}, F_{i}) ∣ i \in I}$ is a semi-open, semi-co-closed dicover with no finite sub-dicover. As in the proof of [2, Theorem 3.5]bd1 we consider the set $F$ of functions f satisfying

dom f is a set of finite subsets of I.

∀ J ∈ dom f, $f (J) = (f_{1} (J), f_{2} (J)) \in P_{J}^{★ ★}$ .

J₁, …, J_n ∈ dom f ⇒ J₁ ∪ … ∪ J_n ∈ dom f.

J, K ∈ dom f, J ⊆ K ⇒ f_l (J) = J ∩ f_l (K),l = 1, 2.

Here $\begin{matrix} P_{J}^{★ ★} & = & {(J_{1}, J_{2}) \in P_{J}^{★} ∣ \forall K finite, \\ J \subseteq K \subseteq I, \exists (K_{1}, K_{2}) \in P_{K}^{★} \\ with J \cap K_{l} = J_{l}, l = 1, 2} . \end{matrix}$ where $\begin{matrix} P_{J} & = {(J_{1}, J_{2}) ∣ J = J_{1} ⊔ J_{2}}, and \\ P_{J}^{★} & = {(J_{1}, J_{2}) \in P_{J} ∣ ⋂_{j \in J_{1}} F_{j} \subseteq ⋁_{j \in J_{2}} G_{j}}, \end{matrix}$ and J = J₁ ⊔ J₂ denotes that {J₁, J₂} is a partition of J. Exactly as in the proof of [2, Theorem 3.5]bd1 it may be verified that $F$ contains an element g satisfying ∪ dom g = I.

Now consider the family $B = {(⋂_{j \in g_{1} (J)} F_{j}, ⋃_{j \in g_{2} (J)} G_{j}) ∣ J \in dom g}$ . It is easy to show that $B$ has the fep. Also ⋂_j∈g₁(J)F_j is semi-closed since each F_j is semi-closed, and likewise ⋃_j∈g₂(J)G_j is semi-open. Hence by (2) we have $⋂_{J \in dom g} (⋂_{j \in g_{1} (J)} F_{j}) \subseteq ⋁_{J \in dom g} (⋃_{j \in g_{2} (J)} G_{j}) .$

Let I₁ = ⋃ {g₁ (J) ∣ J ∈ dom g}, I₂ = I \ I₁. Then (I₁, I₂) is a partition of I, and I₂se ⋃ {g₂ (J) ∣ J ∈ dom g}. This gives us $\begin{matrix} ⋂_{J \in dom g} (⋂_{j \in g_{1} (J)} F_{j}) \\ = ⋂_{i \in I_{1}} F_{i} \subseteq ⋁_{i \in I_{2}} G_{i} \subseteq ⋁_{J \in dom g} (⋃_{j \in g_{2} (J)} G_{j}), \end{matrix}$ which is a contradiction.

(3)implies(1) First take semi-open sets G_i, i ∈ I, with S = ⋁ _i∈IG_i. For i ∈ I let F_i =∅. Then $C = {(G_{i}, F_{i}) ∣ i \in I}$ is a semi-open, semi-co-closed dicover, so has a finite sub-dicover {(G_j, F_j) ∣ j ∈ J}. For the partition J₁ =∅, J₂ = J of J, $S = ⋂_{j \in J_{1}} F_{j} \subseteq ⋃_{j \in J_{2}} G_{j},$ whence S = ⋃ _j∈JG_j, and $(S, S, τ, κ)$ is semi-compact. semi-cocompactness is proved in an analogous way.

To establish semi-stability, let F ≠ S be semi-closed and G_i, i ∈ I, be semi-open sets with Fse ⋁ _i∈IG_i. Define $C = {(S, F)} \cup {(G_{i}, \emptyset) ∣ i \in I}$ . It is clear that $C$ is a semi-open, semi-co-closed dicover, and hence has a finite sub-dicover $C_{1}$ . If $C_{1} = {(G_{j}, \emptyset) ∣ j \in J}$ , J finite, then the fact that $C_{1}$ is a dicover implies ⋃_j∈JG_j = S, whence Fse ⋃ _j∈JG_j. On the other hand, if $(S, F) \in C_{1}$ then we again obtain Fse ⋃ _j∈JG_j, as required. semi-costability can be proved in a similar way.

Hence (τ, κ) is semi-dicompact. □

6 Soft fuzzy topological spaces

In this section, we will give an overview of soft fuzzy sets and topologies and mappings between them by [21].

6.1 Soft fuzzy sets

Firstly, 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 2.2 (3).

Definition 6.1. Let X be a set, μ an $𝕀$ -fuzzy subset of X and M ⊆ X. Then the pair (μ, M) will be called a soft fuzzy subset of X. The set of all soft fuzzy subsets of X will be denoted by SF (X).

Now we consider unit interval texture $(𝕀, I, ι)$ which is given Examples 2.2 (4). If we set $I$ -fuzzy subsets of X $F_{𝕀} (X) = {η ∣ η : X \to I}$ then by [22, Proposition 4.2]tb1 the mapping $η \to A_{η} A_{η} = {(x, t) ∣ t \in η (x)}$ is an isomorphism from $F_{𝕀} (X)$ to the texturing $P (X) \otimes I$ of $X \times 𝕀$ where $(X, P (X), π_{X})$ is complemented discrete texture which is given Examples 2.2 (1).

On the other hand, a mapping from $F_{𝕀} (X)$ to SF (X) is given by η → (η₁, η₂) where $η_{1} (x) = sup η (x) and η_{2} = {x \in X ∣ η_{1} (x) \in η (x)}$

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

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

Definition 6.3. The relation ⊑ on SF (X) is given by (μ, M) ⊑ (ν, N) ⇔ A_ξ(μ,M) ⊆ A_ξ(ν,N).

Proposition 6.4. Let $F = {(μ_{j}, M_{j}) \in SF (X) | \in J}$ .

$F$ has the greatest lower bound, in (SF (X) , ⊑),denoted by ⊓_j∈J (μ_j, M_j) and given by⊓_j∈J (μ_j, M_j) = (μ, M) where μ (x) = ⋀ _j∈Jμ_j (x) ∀x ∈ X and M = {x ∈ X ∣ ∀ j ∈ J, x ∈ M_j or μ (x) < μ_j (x)}.

$F$ has the least upper bound, in (SF (X) , ⊑), denoted by ⨆_j∈J (μ_j, M_j) and given by ⨆_j∈J (μ_j, M_j) = (μ, M) where μ (x) = ⋁ _j∈Jμ_j (x) ∀x ∈ X and M = {x ∈ X ∣ ∃ j ∈ J with x ∈ M_j and μ (x) = μ_j (x)}.

Corollary 6.5. For all (μ_j, M_j) ∈ SF (X), j ∈ J, we have $\begin{matrix} A_{ξ (⊓_{j \in J} (μ_{j}, M_{j}))} = ⋂_{j \in J} A_{ξ (μ_{j}, M_{j})}, and \\ A_{ξ (⊔_{j \in J} (μ_{j}, M_{j}))} = ⋃_{j \in J} A_{ξ (μ_{j}, M_{j})} . \end{matrix}$

Definition 6.6. For (μ, M) ∈ SF (X), the soft fuzzy set (μ, M) ′ = (1 - μ, X \ M) is called the complement of (μ, 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 so, to $P (X) \otimes I$ .

In this study, the product texture space $(X, P (X), σ_{X}) \otimes (𝕀, I, ι)$ will denoted by $(Z_{X}, Z_{X}, z_{X})$ , that is $\begin{matrix} (X, P (X), σ_{X}) \otimes (𝕀, I, ι) \\ = (X \times 𝕀, P (X) \otimes I, σ_{X} \otimes ι) = (Z_{X}, Z_{X}, z_{X}) . \end{matrix}$

By the definition of _sX ⊗ ι, we have $z_{X} (A_{ξ (μ, M)}) = A_{ξ (μ, M)^{'}} = A_{ξ (1 - μ, X \ M)}$ for all (μ, M) ∈ SF (X).

6.2 The mappings between soft fuzzy sets

Now let X, Y be sets and $\overset{ˇ}{:} X \to Y$ a point function. If we denote the identity on $𝕀$ by id then $< \overset{ˇ}{,} id > : Z_{X} \to Z_{Y}$ defined by $< \overset{ˇ}{,} id > (x, s) = (\overset{ˇ}{(} x), s)$ is ω-preserving which is given in Note 2.7, regarded as a mapping from $(Z_{X}, Z_{X})$ to $(Z_{Y}, Z_{Y})$ . Let us denote by SF-Set the construct whose objects are pairs (X, SF (X)) and morphisms from Set. Specially we define $\begin{array}{l} G ((X, S F (X)) \overset{φ}{\to} (Y, S F (Y))) \\ = (Z_{X}, Z_{X}) \overset{< φ, i d >}{\to} (Z_{Y}, Z_{Y}))) \end{array}$

The point function may be used to define mapping between SF (X) to SF (Y). To this end we look at the difunction (f, F) corresponding to $< \overset{ˇ}{,} id > (x, s) = (\overset{ˇ}{(} x), s)$ as in Note 2.7.

In view of the isomorphisms between SF (X) and $Z_{X}$ , and between SF (Y) and $Z_{Y}$ , the (co-) image operator and inverse (co-) image operator are characterized as detailed below.

Proposition 6.7. Let $\overset{ˇ}{:} X \to Y$ be a point function.

The mapping from SF (X) to SF (Y) corresponding to the image operator of the difunction (f, F) is given by $\begin{array}{l} φ^{⇀} = (ν, N) where ν (y) \\ = \sup {μ (x) ∣ y = φ (x)}, and \\ N = {φ (x) ∣ x \in M and ν (φ (x)) = μ (x)} . \end{array}$

The mapping from SF (X) to SF (Y) corresponding to the co-image operator of the difunction (f, F) is given by $\begin{array}{l} φ^{⇁} = (ν, N) where ν (y) \\ = \inf {μ (x) ∣ y = φ (x)}, and \\ N = Y ∖ {φ (x) ∣ x \in X ∖ M and \\ ν (φ (x)) = μ (x)} . \end{array}$

The mapping from SF (Y) to SF (X) corresponding to the inverse image and inverse co-image of the difunction (f, F) is given by $φ^{\leftarrow} (ν, N) = (ν ° φ, φ^{- 1} [N]) .$

As given detailed in this subsection, it is shown that [21] 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.

Soft fuzzy sets are shown to have a richer mathematical theory than classical $𝕀$ -fuzzy sets. Furthermore, soft fuzzy topologies on X are specialized the notion of $𝕃$ -topologies on X.

6.3 Soft fuzzy topological spaces

In this subsection, we recall [21] soft fuzzy topological spaces and some properties with related to ditopologies.

Note that (1, X) and (0, ∅) are the top and bottom element in the soft fuzzy lattice SF (X), respectively.

Definition 6.8. Let X be a set. A subset T ⊆ SF (X) is called an SF-topology on X if

SFT1 (0, ∅) ∈ T and (1, X) ∈ T,

SFT2 (μ_j, M_j) ∈ T, j = 1, 2, …, n $\Rightarrow ⊓_{j = 1}^{n} (μ_{j}, M_{j}) \in T$ ,

SFT2 (μ_j, M_j) ∈ T, j ∈ J ⇒ ⨆ _j∈J (μ_j, M_j) ∈ T.

As usual, the elements of T are called open, and those of T′ = {(μ, M) ∣ (μ, M) ′ ∈ T} closed.

Example 6.9. Let X = {a, b, c, d}. Consider the following soft fuzzy sets (μ_j, M_j) on X, j = 1 ⋯4, where $\begin{array}{l} M_{1} = {a, b}, μ_{1} = {(a, 1), (b, 1), (c, 0), (d, 0)} \\ M_{2} = {b, c}, μ_{2} = {(a, 0.7), (b, 0.6), (c, 0.5), (d, 0.4)} \\ M_{3} = {b}, μ_{3} = {(a, 0.7), (b, 0.6), (c, 0), (d, 0)} \\ M_{4} = {a, b, c}, μ_{4} = {(a, 1), (b, 1), (c, 0.5), (d, 0.4)} \end{array}$

Then T = {(0, ∅) , (1, X) , (M₁, μ₁) , (M₂, μ₂) , (M₃, μ₃) , (M₄, μ₄)} is SF-topology on X.

Definition 6.10. Let T be an SF-topology on X and V an SF-topology on Y. Then a function $\overset{ˇ}{:} X \to Y$ is called T - V continuous if.

SF-topological spaces and continuous functions between the base sets define a construct category which denote by SF-Top.

Now let us 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.

The closure of a soft fuzzy set (μ, M) will be denoted by $\bar{(μ, M)}$ . It is given by $\bar{(μ, M)} = ⊓ {(ν, N) ∣ (μ, M) ⊑ (ν, N) \in T^{'}} .$

Likewise the interior (μ, M) ° of (μ, M) is given by ${(μ, M)}^{\circ} = ⨆ {(ν, N) ∣ (ν, N) \in T, (ν, N) ⊑ (μ, M)} .$

Note 6.11. We observe that $A_{ξ \bar{(μ, M)}}$ , which is just the closure of A_ξ(μ,M) with respect to the ditopology (A_ξ[T], A_ξ[T′]). Likewise, 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 $< \overset{ˇ}{,} id >$ , $\overset{ˇ}{\in} Set (X, Y)$ , will be denoted by SF-Ditop. In [21], it was shown that $G : SF - Top \to SF - Ditop$ 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-compactness presented in the next sections.

7 Semi-open sets in soft fuzzy topological spaces

Proposition 7.1. Let T be an SF-topology on X and (A_ξ[T], A_ξ[T′]) be the corresponding complemented ditopology on $(Z_{X}, Z_{X}, z_{X})$ . Then $A_{ξ (μ, M)} \in Z_{X}$ is

semi-open if and only if $(μ, M) ⊑ \bar{(μ, M)^{\circ}}$ ,

semi-closed if and only if $(\bar{(μ, M)})^{\circ} ⊑ (μ, M)$ .

Proof. By Definition 6.3, we observe that if (μ, M) ⊑ (ν, N), then A_ξ(μ,M) ⊆ A_ξ(ν,N) for all (μ, M) , (ν, N) ∈ SF (X). On the other hand, we have $cl (A_{ξ (μ, M)}) = A_{ξ \bar{(μ, M)}}$ and int (A_ξ(μ,M)) = A_ξ(μ,M)° from Note 6.11. Hence, the proof is immediately obtained, by using these arguments. □

For the complemented ditopologies, by [11, Proposition 2.2]ds1, the notions of semi-openness and semi-closedness coincide. Since the ditopology $(A_{\overset{ˇ}{[} T]}, A_{\overset{ˇ}{[} T^{'}]})$ complemented this will be the case here, we need only consider the semi-openness for SF-topologies, regarding the definition of ditopological semi-closed set as giving an alternative description of the semi-open set. Hence, we make the following definiton:

Definition 7.2. Let T be an SF-topology on X. Then (μ, M) ∈ SF (X) is called 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 called soft fuzzy semi-closed (written SF-semi-closed) if (μ, M) ′ is SF-semi-open.

Proposition 7.3. Let T be an SF-topology on X and (μ, M) ∈ SF (X). Then (μ, M) is SF-semi-open if and only if there exists an open set (ν, N) ∈ T such that $(ν, N) ⊑ (μ, M) ⊑ \bar{(ν, N)}$ .

Proof. Since (μ, M) ° ∈ T for all (μ, M) ∈ SF (X), the proof is elementary and is omitted. □

The proofs of the following implications are obvious and omitted.

Corollary 7.4. Let T be an SF-topology on X. Then:

If (μ, M) ∈ T, then (μ, M) is SF-semi-open.

If (μ, M) is SF-semi-open and $(μ, M) ⊑ (ν, N) ⊑ \bar{(μ, M)}$ , then (ν, N) is also SF-semi-open.

Arbitrary join of SF-semi-open sets is SF-semi-open and arbitrary intersection of SF-semi-closed sets is SF-semi-closed.

The semi-closure of a soft fuzzy set (μ, M) will be denoted by

{\bar{(μ, M)}}^{s}

. It is given by

\begin{matrix} {\bar{(μ, M)}}^{s} = ⊓ {(ν, N) ∣ (μ, M) \\ ⊑ (ν, N), (ν, N) is SF - semi - closed} . \end{matrix}

Likewise the semi-interior (μ, M) ^∘_s of (μ, M) is given by $\begin{matrix} (μ, M)^{\circ_{s}} = ⨆ {(ν, N) ∣ (ν, N) \\ ⊑ (μ, M), (ν, N) is SF - semi - open} . \end{matrix}$

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, $\begin{matrix} (μ, M) ⊑ {\bar{(μ, M)}}^{s} ⊑ \bar{(μ, M)} and \\ (μ, M)^{\circ} ⊑ (μ, M)^{\circ_{s}} ⊑ (μ, M) . \end{matrix}$

Proposition 7.5. Let T be an SF-topology on X and (μ, M) ∈ SF (X). Then $(μ, M)^{\circ_{s}} = (μ, M) ⊓ \bar{(μ, M)^{\circ}}$ .

Proof. We set (μ, M) ^∘_s = (ν, N). By the above corollary, since (μ, M) ^∘_s is semi-open, we have $\bar{(ν, N)^{\circ}} ⊑ (ν, N)$ . Therefore, $(ν, N) ⊑ \bar{(μ, M)^{\circ}}$ , and so $(ν, N) ⊑ (μ, M) ⊓ \bar{(μ, M)^{\circ}}$ . To obtain the other inclusion, we observe that $(μ, M)^{\circ} ⊑ (μ, M) ⊓ \bar{(μ, M)^{\circ}}$ , $(μ, M)^{\circ} ⊑ \bar{((μ, M) ⊓ \bar{(μ, M)^{\circ}})^{\circ}}$ and $(μ, M) ⊓ \bar{(μ, M)^{\circ}} ⊑ \bar{(μ, M)^{\circ}} ⊑ \bar{((μ, M) ⊓ \bar{(μ, M)^{\circ}})^{\circ}}$ . Hence, $(μ, M) ⊓ \bar{(μ, M)^{\circ}}$ is semi-open, and so $(μ, M) ⊓ \bar{(μ, M)^{\circ}} ⊑ (μ, M)^{\circ_{s}} = (ν, N)$ . □

Definition 7.6. Let T be an SF-topology on X and V an SF-topology on Y. Then a function $\overset{ˇ}{:} X \to Y$ is called

semi-continuous if (ν, N) ∈ V, then is SF-semi-open in SF (X),

semi-irresolute if (ν, N) is SF-semi-open in SF(Y), then is SF-semi-open in SF (X).

Proposition 7.7. Let T be an SF-topology on X and V an SF-topology on Y. Let $\overset{ˇ}{:} X \to Y$ be a function.

The following are equivalent:

φ is semi-continuous,

$\begin{array}{l} (φ^{⇁} (μ, M)) ° ⊑ φ^{⇁} (μ, M) °_{s}, \forall (μ, M) \\ \in S F (X), \end{array}$

$\begin{array}{l} φ^{\leftarrow} ((ν, N) °) ⊑ (φ^{\leftarrow} (ν, N)) °_{s}, \forall (ν, N) \\ \in S F (Y) . \end{array}$

The following are equivalent:

φ is semi-irresolute,

$\begin{array}{l} (φ^{⇁} (μ, M)) °_{s} ⊑ φ^{⇁} (μ, M) °_{s}, \forall (μ, M) \\ \in S F (X) . \end{array}$

$\begin{array}{l} φ^{\leftarrow} (ν, N) °_{s} ⊑ (φ^{\leftarrow} (ν, N)) °_{s} . \forall (ν, N) \\ \in S F (Y) . \end{array}$

Proof. By Proposition 6.7, is co-image operator from SF (X) to SF (Y), and is also inverse (co)-image operator from SF (Y) to SF (X). Hence, we omit the proof which follows the same lines as that of [11, Proposition 2.4, and Proposition 2.6]ds1. □

8 Semi-compactness in soft fuzzy topological spaces

Definition 8.1. Let T be an SF-topology on X. Then (μ, M) ∈ SF (X) is called semi-compact if it satisfies either, and hence both, of the following equivalent conditions:

whenever (μ, M) ⊑ ⨆ _j∈J (μ_j, M_j), (μ_j, M_j) ∈ T, j ∈ J, there is finite subset I of J with (μ, M) ⊑ ⨆ _i∈I (μ_i, M_i).

whenever ⊓_j∈J (ν_j, N_j) ⊑ (μ, M), (ν_j, N_j) ∈ T′, j ∈ J, there is finite subset I of J with ⊓_i∈I (ν_i, N_i) ⊑ (μ, M).

Proposition 8.2. Let T be an SF-topology on X and (A_ξ[T], A_ξ[T′]) be the corresponding complemented ditopology on $(Z_{X}, Z_{X}, z_{X})$ . Then (μ, M) ∈ SF (X) is semi-compact if and only if $A_{ξ (μ, M)} \in Z_{X}$ is semi-compact,equivalently semi-cocompact.

Proof. Necessity: Let (μ, M) ∈ SF (X) be semi-compact. Take a family {A_{ξ(μ_j,M_j)} ∈ A_ξ[T] ∣ j ∈ J} such that A_ξ(μ,M) ⊆ ⋁ _j∈JA_{ξ(μ_j,M_j)}. By Corollary 6.5, we have A_ξ(μ,M) ⊆ A_{ξ(⊔_j∈J(μ_j,M_j))}, and so (μ, M) ⊑ ⨆ _j∈J (μ_j, M_j). Since (μ, M) is semi-compact, there is a finite set I ⊆ J such that (μ, M) ⊑ ⨆ _i∈I (μ_i, M_i). By the definition of “⊑”, we may write A_ξ(μ,M) ⊑ A_{ξ(⊔_i∈I(μ_i,M_i))}, and deduce that A_ξ(μ,M) is semi-compact in $Z_{X}$ . Since $(Z_{X}, Z_{X})$ is complemented, A_ξ(μ,M) is also semi-cocompact from Proposition 3.3.

Sufficiency: Let (μ, M) ⊑ ⨆ _j∈J (μ_j, M_j), (μ_j, M_j) ∈ T, j ∈ J. Then A_{ξ(μ_j,M_j)} ∈ A_ξ[T], for all j ∈ J, and A_ξ(μ,M) ⊆ A_{ξ(⊔_j∈J(μ_j,M_j))} = ⋁ _j∈JA_{ξ(μ_j,M_j)}. Since A_ξ(μ,M) is semi-compact there is a finite set I ⊆ J such that A_(μ,M) ⊆ ⋁ _i∈IA_{ξ(μ_i,M_i)}. By using same arguments which in necessity, we have (μ, M) ⊑ ⨆ _i∈I (μ_i, M_i), and so we deduce that (μ, M) is semi-compact. □

Now let T be an SF-topology on X and (μ, M) ∈ SF (X). Recall that [21] if every open cover of (μ, M) has a finite sub-cover, then it is called compact. Hence it is obtained that if (μ, M) is semi-compact, then it is compact by Corollary 7.4 (i).

Proposition 8.3. Let T be an SF-topology on X and V an SF-topology on Y. Let $\overset{ˇ}{:} X \to Y$ be a semi-irresolute function. If (μ, M) ∈ SF (X) is semi-compact then and is semi-compact in SF (Y).

Proof. is image operator and is co-image operator from SF (X) to SF (Y). If we replace f^→ and F^→ with and in Theorem 3.5 and Proposition 3.7 then the proof is automatically obtained. □

Definition 8.4. Let T be an SF-topology on X. Then (μ, M) ∈ SF (X) is called semi-stable if it satisfies either, and hence both, of the following equivalent conditions:

For (μ, M) ∈ T′ \ {(1, ∅)}, whenever (μ, M) ⊑ ⨆ _j∈J (μ_j, M_j), (μ_j, M_j) ∈ T, j ∈ J, there is a finite subset I ⊆ J with (μ, M) ⊑ ⨆ _i∈I (μ_i, M_i).

For (μ, M) ∈ T′ \ {(0, ∅)}, whenever ⊓_j∈J (μ_j, M_j) ⊑ (μ, M), (μ_j, M_j) ∈ T′, j ∈ J,there is a finite subset I ⊆ J with ⊓_i∈I (μ_i, M_i) ⊑ (μ, M).

Proposition 8.5. Let T be an SF-topology on X and (A_ξ[T], A_ξ[T′]) be the corresponding complemented ditopology on $(Z_{X}, Z_{X}, z_{X})$ . Then (μ, M) ∈ SF (X) is semi-stable if and only if $A_{ξ (μ, M)} \in Z_{X}$ is semi-stable, equivalently co-semi-stable.

Proof. When the same arguments are used in the proof’s Proposition 8.2, the proof is clear. □

Proposition 8.6. Let T be an SF-topology on X and V an SF-topology on Y. Let $\overset{ˇ}{:} X \to Y$ be a semi-irresolute function. If (μ, M) ∈ SF (X) is semi-stable then and is semi-stable in SF (Y).

Proof. In the proof of Theorem 4.6, if we take and instead of f^→, f^← respectively, then the proof is obtained. □

9 Conclusion

Breaking the link between the semi-open and semi-closed sets means that certain results in the theory of classical topological spaces cannot hold in the theory of general ditopological texture spaces. For example, while every semi-open cover of a topological space has a finite subcover if and only if every family of semi-closed sets with the finite intersection has a non-empty intersection, this does not hold for ditopologies in general. The first statement is taken as the definition of semi-compactness, and the second as a dual concept called semi-cocompactness. Further thee concepts are equivalent for complemented ditopologies. On the other hand, semi-compact ditopological texture spaces lack many of the nice properties of semi-compact spaces, and the same is true of semi-cocompact spaces. For instance, semi-closed elements of the texturing need not be semi-compact. Hence the dual notions of semi-stability and semi-costability are presented for ditopological texture spaces. It is shown that such spaces have some of the pleasant properties of semi-compact topological spaces. Furthermore, by using the isomorphism between soft fuzzy topological spaces and ditopological texture spaces, the notions of semi-compactness and semi-stability are defined, and some fruitful result are proved in soft fuzzy topological spaces. We hope that these concepts provide useful applications for mathematical approach of uncertainties.

Footnotes

Acknowledgments

The author 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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