Strong β -irresoluteness and weak β -irresoluteness based on continuous valued logic

Abstract

The aim of this paper is to introduce and study the concepts of fuzzifying semipre-θ-neighborhood system of a point, fuzzifying semipre-θ-closure of a set, fuzzifying semipre-θ-interior of a set, fuzzifying semipre-θ-open sets and fuzzifying semipre-θ-closed sets in fuzzifying topological spaces, and investigate some of their basic properties. Also, we present two types of functions in a fuzzifying topological spaces called fuzzifying strong β -irresolute and fuzzifying weak β -irresolute functions. The interrelations of these functions with the parallel existing allied concepts are established. Further, several characterizations of these functions along with different conditions for their existence are obtained.

Keywords

Łukasiewicz logic fuzzifying topology fuzzifying strong semipre-irresolute function

1 Introduction

Many authors (see [1 –14]) were interested in topology-like family (for examples, the family of all β -open sets, and the family of all semipre-θ-open sets), fuzzy topology-like family (for examples, the family of all fuzzy semi-preopen sets, and the family of all fuzzy semipre-θ-open sets), or fuzzifying topology-like mapping (for example, the mapping determining degree of being a fuzzifying β -open set). Actually, research on spaces analogous to topological spaces and mappings analogous to continuous mappings between topological spaces may has certain drive effect on research on theory of rough set, spatial reasoning, knowledge spaces and implicational spaces, and logic (see [15 , 20–30]). For this reason, we will extent the concepts of semipre-θ-neighborhood system of a point, semipre-θ-closure of a set, semipre-θ-interior of a set, semipre-θ-open sets and semipre-θ-closed sets to fuzzifying topological spaces. We will also present the concepts of fuzzifying strong β -irresolute function and fuzzifying weak β -irresolute function 1

1
We take the value lattice as [0, 1], the standard MV-algebra, just for the purpose of readableness. Actually, the results in this paper can be extended to the case of MV-algebras

The rest of the paper is organized as follows. In Section 2 we briefly recall some concepts and results on fuzzifying topology, which are used in the sequel. In Section 3 we define the concepts of fuzzifying semipre-θ-neighborhood system of a point, fuzzifying semipre-θ-closure of a set, fuzzifying semipre-θ-interior of a set, fuzzifying semipre-θ-open set, and fuzzifying semipre-θ-closed set in the framework of fuzzifying topology, and generalize some fundamental results in topological spaces (see Definition 8, Theorem 10 (3), (5), and (6), Theorems 20 and 26). We prove that every fuzzifying semipre-θ-closed set is fuzzifying β -closed set (see Theorem 10(6)). In Section 4 we define the fuzzifying strong β -irresolute functions and fuzzifying weak β -irresolute functions in fuzzifying topological spaces. We also give a list of conditions (see Definitions 30 and 37) to describe fuzzifying strong β -irresolute functions and fuzzifying weak β -irresolute functions (see Theorems 31 and 38). Furthermore, we show that some statements which are true in classical topological spaces need not be true in fuzzifying topological spaces (see Remarks 32 and 34, and Examples 33 and 35). In addition, compositions of fuzzifying weak β -irresolute functions with fuzzifying β -irresoluteness are presented. In the last section, the main results obtained are briefly summarized, and some related topics are addressed for further study.

2 Preliminaries

In this section, we offer some concepts and results in fuzzifying topology, which will be used in the sequel. For the details, we refer to [10, 17]. First, we present a number of fuzzy logical notations and properties.

For any formulas φ, the symbol [φ ] means the truth value of φ, where the set of truth values is the unit interval [0, 1]. A formula φ is valid, we write ⊨φ if and only if [φ ] =1 for every interpretation. Let X be a universe of discourse, and let P (X) and ℱ (X) denote the classes of all crisp subsets and fuzzy subsets of X, respectively. Then, for $\tilde{A}, \tilde{B} \in ℱ (X),$ and any x ∈ X,

(1)[¬ α ] =1 - [α ];

[α ∧ β ] = min ([α ], [β ]);

[α → β ] = min (1, 1 - [α ]+ [β ]);

[∀ xα (x)] =inf_{x ∈X} [α (x)];

$[x \in \tilde{A}] = \tilde{A} (x);$

[∃ xα (x)]: = [¬ (∀ x ¬ α (x))].

(2) [α ∨ β ]: = [¬ (¬ α ∧ ¬ β)];

[α ↔ β ]: = [(α → β) ∧ (β → α)];

$\begin{array}{l} [\tilde{A} \subseteq \tilde{B}] : & = [\forall x (x \in \tilde{A} \to x \in \tilde{B})] \\ = \inf_{x \in X} \min (1, 1 - \tilde{A} (x) + \tilde{B} (x)) : \end{array}$

$[\tilde{A} \equiv \tilde{B}] : = [(\tilde{A} \subseteq \tilde{B}) \land (\tilde{B} \subseteq \tilde{A})];$

$\begin{array}{l} [α \overset{•}{\lor} β] : = [\neg (α \to \neg β)] = \min (1, [α] + [β]); \\ [α \underset{•}{\land} β] : = [\neg α \to β] = \max (0, [α] + [β] - 1) . \end{array}$

Second, we give the following definitions and results in fuzzifying topology which are useful in the rest of the present paper.

Definition 1. (cf. [10]) Let X be a set. If ℱ (P (X)) satisfies the following conditions:

$T$ (X) = $T$ (∅) =1;

For any A, B, $T$ (A ∩ B) ≥ $T$ (A) ∧ $T$ (B);

For any ${A_{λ} : λ \in Λ}$ , $T (\underset{λ \in Λ}{\cup} A_{λ}) \geq \underset{λ \in Λ}{\land} T (A_{λ})$ .

Then $T$ is called a fuzzifying topology, (X, $T$ ) is called a fuzzifying topological space, A ∈ P (X) with $T$ (A) >0 is called a fuzzifying open set (write as A ^∈ $T$ ), and $T$ (A) is called the degree of open of A.

Definition 2. (cf. [10, 17]) Let (X, $T$ ) be a fuzzifying topological space. Then

(1) ℱ (P (X)), defined by $T$ ^c (A) = $T$ (X ∼ A) (∀ A ∈ P (X)), is called a fuzzifying cotopology, B ∈ P (X) with $T$ (X ∼ A) >0 is called a fuzzifying closed set (write as B ^∈ $T$ ^c), and $T$ ^c (A) is called the degree of close of A, where X ∼ A is the complement of A.

(2) ℱ (P (X)), defined by $N_{x} (A) = \underset{x \in B \subseteq A}{\lor} ℐ (B) (\forall A \in P (X))$ is called a fuzzifying neighborhood system of a point x ∈ X, C ∈ P (x) with $N$ _x (C) >0 is called a fuzzifying neighborhood of x (write as C ^∈ $N$ _x), and $N$ _x (C) is called the degree of x being a neighborhood of x.

(3) The fuzzifying cluster operator Cl (A) ∈ ℱ (X) of a set A ⊆ X is defined by Cl (A) (x) =1 - $N$ _x (X ∼ A) (∀ x ∈ X), and Cl (A) (x) is called the degree of x being a cluster point of A.

(4) The fuzzifying interior operator Int (A) ∈ ℱ (X) of a set A ⊆ X is defined by Int (A) (x) = $N$ _x (X ∼ A) (∀ x ∈ X), and Int (A) (x) is called the degree of x being a interior point of A.

Definition 3. (cf. [11]) Let (X, $T$ ) be a fuzzifying topological space. Then

(1) A ∈ P (X) satisfying the following condition is called fuzzifying β -open set (write as A ^∈ $T$ _β): $ℐ_{β} (A) = \underset{x \in A}{\land} C l (I n t (C l (A))) (x) > 0,$

where $T$ _β (A) is called the degree of β -open of A. (2) A ∈ P (X) satisfying $ℐ_{β} = ℐ_{β} (X \sim A) > 0$ is called a fuzzifying β -closed set (write as $A^{\in} T_{β}^{c}$ ), and $T_{β}^{c} (A)$ is called the degree of β -closed of A.

(3) $N_{x}^{β} \in F (P (X))$ , defined by $N_{x}^{β} = \underset{x \in B \subseteq A}{\lor} ℐ_{β} (β) (\forall A \in P (X))$ is called a fuzzifying β -neighborhood system of a point x ∈ X, C ∈ P (X) with $N_{x}^{β} (C) > 0$ is called a fuzzifying β -neighborhood of x (write as $C^{\in} N_{x}^{β}$ ), and $N_{x}^{β} (C)$ is called the degree of C being a β -neighborhood of x.

(4) Cl _β (A) ∈ ℱ (X), defined by $C l_{β} (A) (x) = 1 - N_{x}^{β} (X \sim A) (\forall x \in X),$ is called a fuzzifying β -cluster point operator of a subset A ⊆ X, x ∈ X with Cl _β (A) (x) >0 is called a fuzzifying β -cluster point of A (write as x ^∈Cl _β (A)), and Cl _β (A) (x) is called the degree of x being a fuzzifying β -cluster point of A.

(5) Int _β (A) ∈ ℱ (X), defined by $I n t_{β} (A) (x) = N_{x}^{β} (A) (\forall x \in X),$ is called a fuzzifying β -interior point operator of a subset A ⊆ X, x ∈ X with Int _β (A) (x) >0 is called a fuzzifying β -cluster point of A (write as x ^∈Int _β (A)), and Int _β (A) (x) is called the degree of x being a fuzzifying β -interior point of A.

Definition 4. (cf. [11]) Let (X, $T$ ) and (Y, $U$ ) be two fuzzifying topological spaces. f ∈ Y ^X satisfying the following condition is called fuzzifying β -continuous function (write as f ^∈ $C$ _β): $C_{β} (f) = \underset{U \in P (Y)}{\land} \min (1, 1 - U (U) + ℐ_{β} (f^{- 1} (U))) > 0,$ where $C$ _β (f) is called the degree of f being a β -continuous function.

Definition 5. (cf. [12]) Let (X, $T$ ) and (Y, $U$ ) be two fuzzifying topological spaces. f ∈ Y ^X satisfying the following condition is called fuzzifying β -irresolute function (write as f ^∈ $I$ _β): $ℐ_{β} (f) = \underset{U \in P (Y)}{\land} \min (1, 1 - U_{β} (U) + ℐ_{β} (f^{- 1} (U))) > 0,$

where $I$ _β (f) is called the degree of f being a β -irresolute function.

3 Fuzzifying semipre-θ-closureand fuzzifying semipre-θ-interior

Definition 6. $sp N_{x}^{θ} \in F (P (X))$ , defined by $\begin{array}{l} s p N_{x}^{θ} (A) = & \underset{B \in P (X)}{\lor} \max (0, N_{x}^{β} (B) + \underset{y \notin A}{\land} N_{y}^{β} & (X \sim B) - 1) \\ (\forall A \in P (X)) \end{array}$

is called a fuzzifying semipre-θ-neighborhood system of a point x ∈ X, C ∈ P (X) with $sp N_{x}^{θ} (C) > 0$ is called a fuzzifying semipre-θ-neighborhood of x (write as $C^{\in} sp N_{x}^{θ}$ ), and $sp N_{x}^{θ} (C)$ is called the degree of C being a semipre-θ-neighborhood of x.

Theorem 7. The mapping $s p N^{θ} : X \to ℱ^{N} (P (X))$ (the set of all normal fuzzy subsets of P (X)), defined by $sp N^{θ} (x) = sp N_{x}^{θ} (\forall x \in X),$ has the following properties:

(1) $⊨ A^{\in} s p N_{x}^{θ} \to x \in A (\forall x \in X, \forall A \in P (X));$

(2) $\begin{array}{l} ⊨ A \subseteq B \to & (A^{\in} s p N_{x}^{θ} \to B^{\in} s p N_{x}^{θ}) \\ (\forall x \in X, \forall A, B \in P (X)); \end{array}$

(3) $\begin{array}{l} ⊨ (A^{\in} s p N_{x}^{θ}) \land (B^{\in} & s p N_{x}^{θ}) \leftrightarrow A \cap^{} B^{\in} s p N_{x}^{θ} \\ (\forall x \in X, \forall A, B \in P (X)) . \end{array}$

Proof. (1) If $[A^{\in} sp N_{x}^{θ}] = 0,$ then the result holds. If $[A^{\in} sp N_{x}^{θ}] > 0,$ then $\begin{array}{l} \underset{B \in P (x)}{\lor} \max (0, N_{x}^{β} (B) + \underset{y \notin A}{\land} N_{y}^{β} (X \sim B) - 1) \\ = \underset{B \in P (X)}{\lor} \max (0, N_{x}^{β} (B) + [C l_{β} (B) \subseteq A] - 1) > 0, \end{array}$ and thus there exists a C ∈ P (X) such that $N_{x}^{β} (C) + [{Cl}_{β} (C) \subseteq A] - 1 > 0 .$ From Theorems 4.3(1) and 5.1(1)(c) in [11], we have $[{Cl}_{β} (C) \subseteq A] > 1 - N_{x}^{β} (C) \geq 1 - [x \in C] \geq 1 - {Cl}_{β} (C) (x) .$ Therefore, $[C l_{β} (C) \subseteq A] = \underset{y \notin A}{\land} (1 - C l_{β} (C) (y)) > 1 - C l_{β} (C) (x),$ and so x ∈ A. Otherwise, if x ∉ A, then $\underset{y \notin A}{\land} (1 - C l_{β} (C) (y)) > 1 - C l_{β} (C) (x) .$ This contradicts with the definition of the infimum. So [x ∈ A ] =1. Thus, $[A^{\in} sp N_{x}^{θ}] \leq [x \in A] .$

(2) If [A ⊆ B ] =0, then the result holds. If [A ⊆ B ] =1 and C ∈ P (X),then $\begin{array}{l} [C l_{β} (C) \subseteq B] = \underset{x \notin X - B}{\land} (1 - C l_{β} (C) (x)) \\ \geq \underset{x \notin X \sim A}{\land} (1 - C l_{β} (C) (x)) = [C l_{β} (C) \subseteq A], \end{array}$ and so $N_{x}^{β} (C) + [{Cl}_{β} (C) \subseteq B] - 1 \geq N_{x}^{β} (C) + [{Cl}_{β} (C) \subseteq A] - 1 .$ Thus, $\begin{array}{l} \underset{c \in P (X)}{\lor} \max (0, N_{x}^{β} (C) + \underset{y \notin B}{\land} N_{y}^{β} (X \sim C) - 1) \\ = \underset{c \in P (X)}{\lor} \max (0, N_{x}^{β} (C) + [C l_{β} (C) \subseteq B] - 1) \\ \geq \underset{c \in P (X)}{\lor} \max (0, N_{x}^{β} (C) + [C l_{β} (C) \subseteq A] - 1) \end{array}$

Hence, $sp N_{x}^{θ} (A) \leq sp N_{x}^{θ} (B) .$

(3) Let φ ₁ (resp., φ ₂, φ ₃) be Cl _β (C) ⊆ A (resp., Cl _β (C) ⊆ B, Cl _β (C) ⊆ A ∩ B). Then $\begin{array}{l} [φ_{3}] = \forall x (x \in C l_{β} (c) \to x \in A \cap B) \\ = \underset{x \in X}{\land} \min (1, 1 - C l_{β} (C) (x) + [(A \cap B) (x)]) \\ = \underset{x \notin A \cap B}{\land} (1 - C l_{β} (C) (x)) = \underset{x \in (X ~ A) \cup^{} (X ~ B)}{\land} (1 - C l_{β} (C) (x)) \\ = \underset{x \in X ~ A}{\land} (1 - C l_{β} (C) (x)) \land \underset{x \in X ~ B}{\land} (1 - C l_{β} (C) (x)) \\ = [φ_{1}] \land [φ], [A \cap B^{\in} s p N_{x}^{0}] \\ = \underset{c \in P (X)}{\lor} \max (0, N_{x}^{β} (C) + \underset{y \in A \cap B}{\land} N_{y}^{β} (X ~ C) - 1) \\ = \underset{c \in P (X)}{\lor} \max (0, N_{x}^{β} (C) + [φ_{3}] - 1) . \end{array}$

Thus, $\begin{array}{l} [s p N_{x}^{θ} (A \cap B)] = \underset{C \in P (X)}{\lor} \max (0, N_{x}^{β} (C) + [φ_{3}] - 1) \\ = \underset{C \in P (X)}{\lor} \max (0, N_{x}^{β} (C) + [φ_{1}] \land [φ_{2}] - 1) \\ = \underset{C \in P (X)}{\lor} \max (0, N_{x}^{β} (C) + [φ_{1}] - 1) \land (N_{x}^{β} (C) + [φ_{2}] - 1) \\ = \underset{C \in P (X)}{\lor} (\max (0, N_{x}^{β} (C) + [φ_{1}] - 1) \\ \land \max (0, N_{x}^{β} (C) + [φ_{2}] - 1)) \\ = (\underset{C \in P (X)}{\lor} \max (0, N_{x}^{β} (C) + [φ_{1}] - 1)) \\ \begin{array}{l} \land (\underset{C \in P (X)}{\lor} \max (0, N_{x}^{β} (C) + [φ_{2}] - 1)) \\ = [s p N_{x}^{θ} (A)] \land [s p N_{x}^{θ} (B)] . \end{array} \end{array}$

In a topological space (X, $T$ ), the semipre-θ-closure [4] of subsets of X is to be the set of all x ∈ X such that A ∩ Cl _β (B) ≠ ∅ for every} B ∈ spO (X, $T$ ) with x ∈ B. We generalize this notion to the fuzzifying setting in the following definition:

Definition 8. The fuzzifying semipre-θ-closure of a set A ⊆ X, define spCl _θ (A) ∈ ℱ (P (X)) as $\begin{array}{l} s p C l_{θ} (A) (x) = \underset{B \in P (X)}{\land} \min (1, 1 - N_{x}^{β} (B) & + \underset{y \in A}{\lor} C l_{β} (B) (y)) \\ (\forall A \in P (X)), \end{array}$ where spCl _θ (A) (x) is called the degree of x being a β -cluster point of A.

Lemma 9. Let (X, $T$ ) be a fuzzifying topological space. Then $⊨ x^{\in} {spCl}_{θ} (A) : = (\forall B) (B^{\in} N_{x}^{β} \to \neg (A \cap {Cl}_{β} (B) \equiv \emptyset)) .$

Proof. Let φ ₁ (resp., φ ₂, φ ₂) be $1 - N_{x}^{β} (B)$ (resp., Cl _β (B), A ∩ Cl _β (B)). By Definition 8,

\begin{array}{l} [\underset{B \in P (X)}{\land} \min (1, [φ_{1}] + \underset{y \in a}{\lor} [φ_{2}] (y))] \\ = & \underset{B \in P (X)}{\land} \min (1, [φ_{1}] + \underset{y \in a}{\lor} \min (A (y), [φ_{2}] (y))) \\ = & \underset{B \in P (X)}{\land} \min (1, [φ_{1}] + \underset{y \in a}{\lor} \max (0, [φ_{3}] (y)))) \\ = & \underset{B \in P (X)}{\land} \min (1, [φ_{1}] + 1 - \underset{y \in a}{\land} \min (1, 1 - \\ [φ_{3}] (y) + 0))) \\ = & \underset{B \in P (X)}{\land} \min (1, [φ_{1}] + 1 - ([φ_{3}] \subseteq \emptyset] \land [\emptyset \subseteq [φ_{3}])) \\ = & \underset{B \in P (X)}{\land} \min (1, [φ_{1}] + 1 - ([φ_{3}] \equiv \emptyset)) \\ = & \underset{B \in P (X)}{\land} \min (1, [φ_{1}] + [(\neg [φ_{3}] \equiv \emptyset)]) \\ = & [(\forall B) (N_{x}^{β} (B) \to \neg (A \cap C l_{β} (B) \equiv \emptyset))] \end{array}

Theorem 10. Let (X, $T$ ) be a fuzzifying topological space. Then

(1) ${spCl}_{θ} (A) (x) = 1 - sp N_{x}^{θ} (X \sim A);$

(2) ⊨spCl _θ (∅) ≡ ∅;

(3) ⊨A ⊆ spCl _θ (A);

(4) $⊨ x \in {spCl}_{θ} (A) \leftrightarrow (\forall B) (B^{\in} sp N_{x}^{θ} \to A \cap B \neq \emptyset);$

(5) ⊨A ⊆ B → spCl _θ (A) ⊆ spCl _θ (B);

(6) ⊨Cl _β (A)⊆ spCl _θ (A);

(7) ⊨spCl _θ (A ∪ B) ≡ spCl _θ (A) ∪ spCl _θ (B);

Proof. (1) spCl _θ (A) (x) $= 1 - sp N_{x}^{θ} (X \sim A)$ . $\begin{array}{l} = \underset{B \in P (X)}{\land} \min (1, 1 - N_{x}^{β} (B) \underset{y \in A}{\lor} C l_{β} (B) (y)) \\ = 1 - \underset{B \in P (X)}{\lor} \max (0, N_{x}^{β} (b) - \underset{y \in A}{\lor} C l_{β} (B) (y)) \\ = 1 - \underset{B \in P (X)}{\lor} \max (0, N_{x}^{β} (B) + 1 - \underset{y \in A}{\lor} C l_{β} (B) (y) - 1) \\ = 1 - \underset{B \in P (X)}{\lor} \max (0, N_{x}^{β} (B) + \underset{y \in A}{\land} (1 - C l_{β} (B) (y)) - 1) \\ = 1 - [(\exists B) ((B \in N_{x}^{β}) \underset{\cdot}{\land} (C l_{β} (B) \subseteq X ~ A))] \\ = 1 - s p N_{x}^{θ} (X ~ A) . \end{array}$

(2) From (1) and the fact that $sp N_{x}^{θ}$ is normal, we have spCl _θ (∅) (x) $= 1 - sp N_{x}^{θ} (X \sim \emptyset) = 0 .$

(3) Let A ∈ P (X) and x ∈ X. Then $sp N_{x}^{θ} (A) = 0$ if x ∉ A; ${spCl}_{θ} (A) (x) = 1 - sp N_{x}^{θ} (X \sim A) = 1 - 0 = 1$ if x ∈ A. Thus [A ⊆ spCl _θ (A)] = 1.

(4) $[(\forall B) (B^{\in} s p N_{x}^{β} \to A \cap B \neq \emptyset)] = \underset{B \subseteq X ~ A}{\land} (1 - s p N_{x}^{β} (B)) = 1 - s p N_{x}^{θ} (X \sim A) = [x \in s p C l_{θ} (A)] .$

(5) If [A ⊆ B ] = 0, then the conclusion holds. If [A ⊆ B ] = 1, then $s p C l_{θ} (A) (x) = 1 - s p N_{x}^{θ} (X ~ A) \leq 1 - s p N_{x}^{θ} (X ~ B) = s p C l_{θ} (B) (x)$ .

(6) From Theorem 5.1 (1) (c) and (d) in [11], we have $\begin{array}{l} C l_{β} (A) (x) & = \underset{B \in P (X)}{\land} \min (1, 1 - N_{x}^{β} (B) + \underset{y \in A}{\lor} (B) (y)) \\ \leq \underset{B \in P (X)}{\land} \min (1, 1 - N_{x}^{β} (B) + \underset{y \in A}{\lor} C l_{β} (B) (y)) \\ = s p C l_{θ} (A) (x) \end{array}$

(7) It is easy to see spCl_θ (A) ∪ spCl_θ (B) ⊆ spCl_θ (A ∪ B). Conversely, let φ ₁ be $1 - N_{x}^{β} (C)$ , and x ∈ X. Then $\begin{array}{l} s p C l_{θ} (A \cup B) (x) \\ = \underset{C \in P (X)}{\land} \min (1, [φ_{1}] + \underset{y \in (A \cup B)}{\lor} C l_{β} (C) (y)) \\ = \underset{C \in P (X)}{\land} \min (1, [φ_{1}] + (\underset{y \in A}{\lor} C l_{β} (C) (y) \lor \underset{y \in B}{\lor} C l_{β} (C) (y))) \\ \leq \underset{C \in P (X)}{\land} \min (1, [φ_{1}] + \underset{y \in A}{\lor} C l_{β} (C) (y)) \\ \lor \underset{C \in P (X)}{\land} \min (1, [φ_{1}] + \underset{y \in B}{\lor} C l_{β} (C) (y)) \\ = s p C l_{θ} (A) (x) \lor s p C l_{θ} (B) (x) . \end{array}$

Remark 11. The equality in Theorem 10 (6) need not be true in general as shown by the following example.

Example 12. Let X = {x, y} and $T$ be a fuzzi-fying topology on X defined by, $T (X) = T (\emptyset) = T ({x}) = 1$ and $T ({y}) = 0$ . Then spCl_θ ({x}) (x) = 0 < Cl_ß ({x})(x) = 1.

Lemma 13. $⊨ A^{\in} s p N_{x}^{θ} \to A^{\in} N_{x}^{β}$ .

Proof. It follows from Theorem 10(6).

Remark 14. The equality in above Lemma need not be true in general as shown by the following example.

Example 15. Let X = {x, y} and $T$ be a fuzzifying topology on X defined in Example 12. Then $N_{x}^{β} ({y}) = 0 < sp N_{x}^{θ} ({y}) = 1 .$

Definition 16. Let (X, $T$ ) be a fuzzifying topological space. then

(1) A ∈ P (X) with $T s p_{θ} (A) = \underset{x \in X ~ A}{\land} (1 - s p c l_{θ} (A) (x)) > 0$ is called a fuzzifying semipre-θ-closed set (write as $A^{\in} T_{sp θ}^{c}$ ) and $T_{sp θ}^{c} (A)$ is called the degree of semipre-θ-closed of A.

(2) A ∈ P (X) with $T_{sp θ} (A) = T_{sp θ}^{c} (X \sim A) > 0$ is called a fuzzifying semipre-θ-open set (write as A ^∈ $T$ _spθ) and $T$ _spθ (A) is called the degree of semipre-θ-open of A.

(3) $sp N_{x}^{θ} \in F (P (X))$ , defined by $s p N_{x}^{θ} (A) = \underset{x \in B \subseteq A}{\lor} J s p_{θ} (B) (\forall A \in P (X))$

Definition 17. The fuzzifying semipre-θ-interior operator spInt _θ (A) ∈ℱ (X) of a set A ⊆ X is defined by ${spInt}_{θ} (A) (x) = sp N_{x}^{θ} (A) (\forall x \in X)$ , where spInt _θ (A) (x) is called the degree of x being a semipre-θ-interior point of A.

Theorem 18. Let (X, $T$ ) be a fuzzifying topological space. then

(1) ⊨spInt _θ (A)≡ X ∼ spCl _θ (X ∼ A);

(2) ⊨spInt _θ (X)≡ X;

(3) ⊨spInt _θ (A)⊆ A;

(4) $⊨ A^{\in} T_{sp θ} \leftrightarrow \forall x (x \in A \to A^{\in} {spN}_{x}^{θ});$

(5) ⊨ (A ^∈ $T$ _spθ)∧ (A ⊆ B) → A ⊆ spInt _θ (B);

(6) ⊨A ⊆ B → spInt _θ (A) ⊆ spInt _θ (B);

(7) ⊨spInt _θ (A)⊆ Int _β (A);

(8) ⊨spInt _θ (A ∩ B) ≡ spInt _θ (A) ∩ spInt _θ (B).

Proof. (1) From Theorem 10(1), we have ${spCl}_{θ} (X \sim A) (x) = 1 - {spN}_{x}^{θ} (A) = 1 - {spInt}_{θ} (A) (x) .$ Then, [spInt _θ (A) ≡ X ∼ spCl _θ (X ∼ A)] =1.

(2) By (1), spInt _θ (X) = X ∼ spCl _θ (X ∼ X) = X ∼ spCl _θ (∅) = X ∼ ∅ = X.

(3) By Theorem 10(3), spInt _θ (A) ≡ X ∼ spCl _θ (X ∼ A) ⊆ X ∼ (X ∼ A) ≡ A.

(4) [A ^∈ $T$ _spθ] $\begin{array}{l} = \underset{x \in A}{\land} (1 - s p C l_{θ} (X ~ A) (x)) \\ = \underset{x \in A}{\land} (1 - \underset{B \in P (X)}{\land} \min (1, 1 - N_{x}^{β} (B) + \underset{y \in X ~ A}{\lor} C l_{β} (B) (y))) \\ = \underset{x \in A}{\land} \underset{B \in P (X)}{\lor} \max (0, N_{x}^{β} (B) + 1 - \underset{y \in X ~ A}{\lor} C l_{β} (B) (y) - 1) \\ = \underset{x \in A}{\land} \underset{B \in P (X)}{\lor} \max (0, N_{x}^{β} (B) + \underset{y \in X ~ A}{\land} (1 - C l_{β} (B) (y) - 1)) \\ = \underset{x \in A}{\land} \underset{B \in P (X)}{\lor} (N_{x}^{β} (B) \underset{\cdot}{\land} \underset{y \in X ~ A}{\lor} (C l_{β} (B) (y))) \\ = [\forall x (x \in A \to \exists B ((B \in N_{x}^{β}) \underset{•}{\land} (C l_{β} (B) \subseteq A)))] \\ = [\forall x (x \in A \to A \in s p N_{x}^{θ})] . \end{array}$

(5) If [A ⊆ B ] =0, then the conclusion holds. If [A ⊆ B ] =1, then $\begin{array}{l} [A \subseteq s p I n t_{θ} (B) = \underset{x \in A}{\land} s p I n t_{θ} (B) (x)] \\ = \underset{x \in A}{\land} s p N_{x}^{θ} (B) \geq \underset{x \in A}{\land} s p N_{x}^{θ} (A) \\ = T s p_{θ} (A) = [(A^{\in} T s p_{θ}) \land (A \subseteq B)] \end{array}$ by (4) and Theorem 7(2).

(6) It is similar to the proof of Theorem 10 (5).

(7) By Lemma 13, ${spInt}_{θ} (A) (x) = sp N_{x}^{θ} (A) \leq N_{x}^{β} (A) = {Int}_{β} (A) (x) .$

(8) By Theorem 10(7), spInt _θ (A ∩ B) (x) =1 - spCl _θ (X ∼ (A ∩ B)) (x) =1 - spCl _θ ((X ∼ A) ∪ (X ∼ B)) (x) =1 - (spCl _θ (X ∼ A) (x) ∪ spCl _θ (X ∼B) (x)) = (1 - spCl _θ (X ∼ A) (x)) ∩ (1 - spCl _θ (X ∼ B) (x)) = spInt _θ (A) (x) ∩ spInt _θ (B) (x).

Remark 19. The equality in Theorem 18(7) need not be true in general as shown in Example 15.

Theorem 20. Let (X, $T$ ) be a fuzzifying topological space. then

(1) ⊨A ^∈ $T$ _spθ↔ A ≡ spInt _θ (A);

(2) $⊨ A^{\in} T_{sp θ}^{c} \leftrightarrow A \equiv {spCl}_{θ} (A) .$

Proof. (1) $[A^{\in} T_{sp θ}] = [X \sim A^{\in} T_{sp θ}^{c}]$ $\begin{array}{l} = \underset{x \in X ~ (X ~ A)}{\land} (1 - s p C l_{θ} (X ~ A) (x)) \\ = \underset{x \in A}{\land} (s p I n t_{θ} (A) (x)) = [A \subseteq s p I n t_{θ} (A)] \\ = ([A \subseteq s p I n t_{θ} (A)] \land [s p I n t_{θ} (A) \subseteq A]) = [A \equiv \\ s p I n t_{θ} (A)] \end{array}$ .

(2) The proof is similar to (1).

Theorem 21. Let (X, $T$ ) be a fuzzifying topological space. Then $T$ _spθ has the following properties:

(1) $T$ _spθ (X) = $T$ _spθ (∅) =1;

(2) For any A, B, $T$ _spθ (A ∩ B) ≥ $T$ _spθ (A) ∧ $T$ _spθ (B);

(3) For any $\begin{array}{l} {A_{λ} : λ \in Λ}, T s p_{θ} (\underset{λ \in Λ}{\cup} A λ) \geq \underset{λ \in Λ}{\land} \\ T s p_{θ} (A_{λ}) . \end{array}$

Proof. (1) Straightforward.

(2) By Theorem 18(4) and Theorem 7(3), $\begin{array}{l} T s p_{θ} (A) \land T s p_{θ} (B) = \underset{x \in A}{\land} s p I n t_{θ} (A) (x) \land \\ \underset{x \in B}{\land} s p I n t_{θ} (B) (x) \\ \leq \underset{x \in A \cap B}{\land} (s p I n t_{θ} (A) (x) \land s p I n t_{θ} (B) (x)) \\ = \underset{x \in A \cap B}{\land} s p I n t_{θ} (A \cap B) (x) = T s p_{θ} (A \cap B) . \end{array}$

(3) By Theorem 18(4), $\begin{array}{l} T s p_{θ} (\cup_{λ \in Λ} A_{λ}) = \underset{x \in \cup_{λ \in Λ} A_{λ}}{\land} s p I n t_{θ} (\underset{λ \in Λ}{\cup} A_{λ}) (x) \\ = \underset{x \in λ}{\land} \underset{x \in A_{λ}}{\land} s p I n t_{θ} (\underset{λ \in Λ}{\cup} A_{λ}) (x) \\ \geq \underset{x \in λ}{\land} \underset{x \in A_{λ}}{\land} s p I n t_{θ} (A_{λ}) (x) = \underset{x \in λ}{\land} T s p_{θ} (A_{λ}) . \end{array}$

Theorem 22. Let (X, $T$ ) be a fuzzifying topological space. then

(1) ⊨A ^∈ $T$ _spθ ↔ ∀ x (x ∈ A → ∃ B (B ^∈ $T$ _spθ ∧ x ∈ B ⊆ A));

(2) $⊨ A^{\in} T_{sp θ} \leftrightarrow \forall x (x \in A \to \exists B (B^{\in} {spN}_{x}^{θ} \land B \subseteq A)) .$

Proof. (1) On the one hand, we have $\begin{array}{l} [\forall x (x \in A \to \exists B (B^{\in} T_{s p θ} \land x \in B \subseteq A))] = \\ \underset{x \in A}{\land} \underset{x \in B \subseteq A}{\lor} T_{s p θ} (B) \end{array}$ and $\underset{x \in A}{\land} \underset{x \in B \subseteq A}{\lor} T_{s p θ} (B) \geq T_{s p θ} (A) .$ On the other hand, let $B$ _x = {B: x ∈ B ⊆ A}. Then, for any $f \in \prod_{x \in A} ℬ_{x}$ , we have $\underset{x \in A}{\cup} f (x) = A$ , $T_{s p θ} (A) = T_{s p θ} (\underset{x \in A}{\cup} f (x)) \geq \underset{x \in A}{\land} T_{s p θ} (f (x))$ and so $\begin{array}{l} T_{s p θ} (A) \geq \underset{f \in ∐_{x \in A} ℬ x}{\lor} \underset{x \in A}{\land} T_{s p θ} (f (x)) = \\ \underset{x \in A}{\land} \underset{x \in B \subseteq A}{\lor} T_{s p θ} (B) \end{array}$ (2) By (1), $\begin{array}{l} [\forall x (x \in A \to \exists B (B^{\in} s p N_{x}^{θ} \land B \subseteq A))] = \\ \underset{x \in A}{\land} \underset{B \subseteq A}{\lor} s p N_{x}^{θ} (B) \\ \underset{x \in A}{\land} \underset{B \subseteq A}{\lor} \underset{x \in C \subseteq A}{\lor} T_{s p θ} (C) = \underset{B \subseteq A}{\lor} \underset{x \in C \subseteq A}{\lor} T_{s p θ} (C) \\ [A^{\in} T_{s p θ}] \end{array}$

Lemma 23. Let (X, $T$ ) be a fuzzifying topological space. then

(1) ⊨ $T$ _spθ⊆ $T$ _β;

(2) $⊨ T_{sp θ}^{c} \subseteq T_{β}^{c} .$

Proof. (1) By Corollary 4.1 [11] and Theorem 10 (6), $\begin{array}{l} [A^{\in} T_{s p θ}] = \underset{x \in A}{\land} (1 - s p C l_{θ} (X ~ A) (x)) \\ \leq \underset{x \in A}{\land} (1 - C l_{β} (X ~ A) (x)) \\ = \underset{x \in A}{\land} (1 - 1 + N_{x}^{β} (A)) = \underset{x \in A}{\land} s p N_{x}^{β} (A) = [A^{\in} T_{β}] . \end{array}$ (2) It follows from (1).

Remark 24. The equality in above Lemma need not be true in general as shown by the following example.

Example 25. Let X = {x, y, z}, and $T$ be a fuzzifying topology on X defined as $T$ (X) = $T$ (∅) = $T$ ({x}) = $T$ ({x, z}) =1, $T$ ({y}) = $T$ ({x, y}) =0, and $T ({z}) = T ({y, z}) = \frac{1}{8} .$ Then $T$ _β ({y}) =0 < $T$ _spθ ({y}) =1, and $T_{β}^{c} ({x, z}) = 0 < T_{sp θ}^{c} ({x, z}) = 1 .$

The following theorem generalizes the well known result in general topological spaces ([4, Theorem 3.5 (1)]).

Theorem 26. ⊨A ^∈ $T$ _β → (spCl _θ (A) ≡ Cl _β (A)).

Proof. First we prove $⊨ [A^{\in} T_{β}] \to [\neg (A \cap {Cl}_{β} (B) \equiv \emptyset) \to \neg (A \cap B \equiv \emptyset)] .$

In fact, by [Corollary 4.1, 11] and [Corollary 5.4(f), 14], $\begin{array}{l} T_{β} (A) \underset{•}{\land} [\neg (A \cap C l_{β} (B) \equiv \emptyset)] \\ = \max (0, \underset{x \in A}{\land} N_{x}^{β} (A) + \underset{y \in A}{\lor} C l_{β} (B) (y) - 1) \\ = \max (0, \underset{y \in A}{\lor} C l_{β} (B) (y) - \underset{x \in A}{\lor} (1 - N_{x}^{β} (A))) \\ \leq \max (0, \underset{x \in A}{\lor} C l_{β} (B) (x) - 1 + N_{x}^{β} (A)) \\ = \max (0 \underset{x \in A}{\lor} ((B \cup D_{β} (B)) (x) - 1 + N_{x}^{β} (A))) \\ = \max (0, \underset{x \in A}{\lor} (\max (B (x), D_{β} (B) (x)) - 1 + N_{x}^{β} (A))) . \end{array}$

If x ∈ B, then $\begin{array}{l} T_{β} (A) \underset{•}{\land} [\neg (A \cap C l_{β} (B) \equiv \emptyset)] \\ = \max (0, \underset{x \in A}{\lor} (B (x) - 1 + N_{x}^{β} (A))) \leq \\ \underset{x \in A}{\lor} N_{x}^{β} (A) (x) \\ \leq \underset{x \in A}{\lor} A (x) \leq \underset{x \in A}{\lor} (A (x) \land B (x)) = [\neg (A \cap B \equiv \\ \emptyset)] \end{array}$

If x ∉ B, then $\begin{array}{l} T_{β} (A) \underset{•}{\land} [\neg (A \cap C l_{β} (B) \equiv \emptyset)] \\ = \max (0, \underset{x \in A}{\lor} (B (x) - 1 + N_{x}^{β} (A))) \\ \begin{array}{l} = \max (0, & \underset{x \in A}{\lor} (\underset{C \in P (X)}{\land} \min \geq (1, 1 - N_{x}^{β} (C) \\ + \underset{y \in A}{\lor} (C ~ {x}) (x) - 1 + N_{x}^{β} (A)))) \end{array} \\ \leq \max (0, \underset{x \in A}{\lor} (1 - N_{x}^{β} (C) + \underset{y \in A}{\lor} (B ~ \\ {x}) ((((y) - 1 + N_{x}^{β} (A)))) \\ = \underset{x \in A}{\lor} (B ~ {x}) (y) = \underset{y \in A}{\lor} B (y) = [\neg (A \cap B \equiv \\ \emptyset)] . \end{array}$

So, $[\neg (A \cap C l_{β} (B) \equiv \emptyset) \to \neg (A \cap (B) \equiv \emptyset)] \geq [T_{β} (A)]$ Therefore,

[spCl _θ (A) ≡ Cl _β (A)]

= [spCl _θ (A) ⊆ Cl _β (A)] ∧ [Cl _β (A) ⊆ spCl _θ (A)]

= [spCl _θ (A) ⊆ Cl _β (A)]

$\begin{array}{l} = (\forall B) (B^{\in} N_{x}^{β} \to \neg (A \cap^{} C l_{β} (B) \equiv \emptyset)) \to \\ (\forall C) (C^{\in} N_{x}^{β} \to \neg (A \cap^{} C \equiv \emptyset)) \\ \geq (\forall B) (B^{\in} N_{x}^{β} \to \neg (A \cap^{} C l_{β} (B) \equiv \emptyset) \to \\ (B^{\in} N_{x}^{β} \to \neg (A \cap^{} B \equiv \emptyset))) \\ = (\forall B) (\neg (A \cap C l_{β} (B) \to \neg (A \cap B \equiv \emptyset)) \\ \geq \underset{B \in P (X)}{\land} T_{β} (A) = T_{β} (A) \end{array}$

Corollary 27. ⊨A ^∈ $T$ → (spCl _θ (A) ≡ Cl _β (A)).

Theorem 28. $⊨ A \in T_{sp θ}^{c} \to ({spCl}_{θ} (A) \equiv {Cl}_{β} (A)) .$

Proof. $\begin{array}{l} [s p C l_{θ} (A) \equiv C l_{β} (A)] \\ = \underset{x \in X}{\land} \min (1, 1 - s p C l_{θ} (A) (x) + C l_{β} (A) (x)) \\ = \underset{x \in X}{\land} \min (1, 1 - s p C l_{θ} (A) (x) + 1 - N_{x}^{β} (X ~ A)) \\ = \underset{x \in X ~ A}{\land} \min (1, 1 - s p C l_{θ} (A) (x) + 1 - N_{x}^{β} (X ~ A)) \\ \geq \underset{x \in X ~ A}{\land} \min (1, 1 - s p C l_{θ} (A) (x)) \\ = \underset{x \in X ~ A}{\land} (1 - s p C l_{θ} (A) (x)) = [A \in ℱ s p_{θ}] . \end{array}$

4 Fuzzifying strong β -irresolute and fuzzifying weak β -irresolute functions

The purpose of this section is to introduce and study the concepts of fuzzifying strong β -irresolute functions and fuzzifying weak β -irresolute functions in fuzzifying topological spaces.

Definition 29. Let (X, $T$ ) and (Y, $U$ ) be two fuzzifying topological spaces. f ∈ Y ^X satisfying the following condition is called fuzzifying strong β -irresolute function (write as f ^∈ $I$ _sβ): $\begin{array}{l} ℐ_{s β} (f) = \underset{U \in P (Y)}{\land} \min (1, 1 - U_{β} (U) + ℐ_{s p^{θ}} (f^{- 1}) (U)) \\ > 0, \end{array}$ where $I$ _sβ (f) is called the degree of f being a strong-β -irresolute function.

Definition 30. Let (X, $T$ ) and (Y, $U$ ) be two fuzzifying topological spaces, f, and consider the following appointments:

(1) α ₁ (f): = (∀ $B) (B^{\in} (T_{β}^{c})^{Y} \to (f$ $- 1 (B))^{\in} {(T_{s p θ}^{c})}^{X}),$ where $(T_{β}^{c})^{Y}$ is the set of all fuzzifying β -closed subset of Y and $(T_{sp θ}^{c})^{X}$ is the set of all fuzzifying semipre-θ-closed subset of X;

(2) α ₂ (f): = (∀ x) (∀ $U) (U^{\in} N_{f (x)}^{β} \to (f$ ^{- 1} (U)) ^∈sp $N_{x}^{θ}),$ where $N_{f (x)}^{β}$ is the fuzzifying β -neighborhood system of f (x) of Y and $sp N_{x}^{θ}$ is the fuzzifying semipre-θ-neighborhood system of x of X;

(3) α ₃ (f): = (∀ x) (∀ $U) (U^{\in} N_{f (x)}^{β} \to (\exists V) ((f$ $(V) \subseteq U) \land (V^{\in} sp N_{x}^{θ})));$

(4) α ₄ (f): = (∀ A) (f $({spCl}_{θ}^{X} (A)) \subseteq {Cl}_{β}^{Y} (f$ (A)));

(5) α ₅ (f): = (∀ $B) ({spCl}_{θ}^{X} (f$ ^{- 1} (B)) ⊆ f $- 1 (C l_{β}^{Y}$ (B)));

(6) α ₆ (f): = (∀ A) (f ^{- 1} (Int _β (A)) ⊆ spInt _θ (f ^{- 1} (A))).

Theorem 31. ⊨f ∈ $I$ _sβ ↔ f ∈ α _k, k = 1, …, 6.

Proof. (1) We prove ⊨f ∈ $I$ _sβ ↔ f ∈ α ₁. $\begin{array}{l} \begin{array}{l} [f \in α_{1}] \\ = & \underset{B \in P (Y)}{\land} \min (1, 1 - {(ℱ_{β})}^{Y} (B) + {(ℱ_{s p θ})}^{X} (f^{- 1} (B))) \\ = & \underset{B \in P (Y)}{\land} \min (1, 1 - U_{β} (Y ~ B) + ℱ_{s p θ} (X ~ f^{- 1} (B))) \\ = & \underset{B \in P (Y)}{\land} \min (1, 1 - U_{β} (Y ~ B) + T_{s p θ} (f^{- 1} (Y ~ B))) \\ = & \underset{U \in P (Y)}{\land} \min (1, 1 - U_{β} (U) + T_{s p θ} (f^{- 1} (U))) \end{array} \\ \begin{array}{l} = & [ℐ s_{β} (f)] . \end{array} \end{array}$

(2) We prove ⊨f ^∈ $I$ _sβ ↔ f ∈ α ₂. First, we show [f ∈ α ₂] ≥ [f ^∈ $I$ _sβ]. If $N_{f (x)}^{β} (U) \leq sp N_{x}^{θ} (f$ ^{- 1} (U)), then

min $(1, 1 - N_{f (x)}^{β} (U) + sp N_{x}^{θ} (f$ ^{- 1} (U))) =1 ≥ [f ^∈ $I$ _sβ].

Suppose $N_{f (x)}^{β} (U) > s p N_{x}^{θ} (f^{- 1} (U)) .$ Then x ∈ f ^{- 1} (A) ⊆ f ^{- 1} (U) if f (x) ∈ A ⊆ U. Thus $\begin{array}{l} N_{f (x)}^{β} (U) > s p N_{x}^{θ} (f^{- 1} (U)) \\ = \underset{f (x) \in A \subseteq U}{\lor} U_{β} (A) - \underset{x \in B \subseteq f^{- 1} (U)}{\lor} ℐ s p_{θ} (A) \\ \leq \underset{f (x) \in A \subseteq U}{\lor} U_{β} (A) - \underset{x \in B \subseteq f^{- 1} (U)}{\lor} ℐ s p_{θ} (f^{- 1} (A)) \\ \leq \underset{f (x) \in A \subseteq U}{\lor} (U_{β} (A) - ℐ s p_{θ} (f^{- 1} (A))) \end{array}$

and $\begin{array}{l} 1 - N_{f (x)}^{β} (U) + s p N_{x}^{θ} (f^{- 1} (U)) \\ \geq \underset{f (x) \in A \subseteq U}{\lor} (1 - U_{β} (A) + ℐ s p_{θ} (f^{- 1} (A))) . \end{array}$

Therefore, $\begin{array}{l} \min (1, 1 - N_{f (x)}^{β} (U) + s p N_{x}^{θ} (f^{- 1} (U))) \\ \geq \underset{f (x) \in A \subseteq U}{\land} (1 - U_{β} (A) + ℐ s p_{θ} (f^{- 1} (A))) \\ \geq \underset{V \in P (Y)}{\land} \min (1 - U_{β} (V) + ℐ s p_{θ} (f^{- 1} (V))) = [f^{\in} ℐ s_{β}], \end{array}$

and $\underset{x \in X}{\land} \underset{U \in P (Y)}{\land} \min (1 - U_{β} (U) + ℐ s p_{θ} (f^{- 1} (U))) = [f^{\in} ℐ s_{β}],$

Second, we show [f ^∈ $I$ _sβ] ≥ [f ∈ α ₂]. By [11, Corollary 4.1] and Theorem 18(4), $\begin{array}{l} [f^{\in} ℐ s_{β}] \\ = \underset{U \in P (Y)}{\land} \min (1, 1 - U_{β} (U) + ℐ s p_{θ} (f^{- 1} (U))) \\ \geq \underset{U \in P (Y)}{\land} \min (1, 1 - \underset{f (x) \in U}{\land} N_{f (x)}^{β} (U) + \\ \underset{x \in f^{- 1} (U)}{\land} s p N_{x}^{θ} (f^{- 1} (U))) \\ \geq \underset{U \in P (Y)}{\land} \min (1, 1 - \underset{x \in f^{- 1} (U)}{\land} N_{f (x)}^{β} (U) + \\ \underset{x \in f^{- 1} (U)}{\land} s p N_{x}^{θ} (f^{- 1} (U))) \\ \geq \underset{x \in X}{\land} \underset{U \in P (Y)}{\land} \min (1 - N_{f (x)}^{β} (U) + s p N_{x}^{θ} (f^{- 1} (U))) \\ = [f \in α_{2}] \end{array}$

(3) We prove ⊨f ∈ α ₂ ↔ f ∈ α ₃. By Theorem 7 (2), $\underset{V \in P (X), f (V) \subseteq U}{\lor} s p N_{x}^{θ} (V) = \underset{V \in P (X), V \subseteq f^{- 1} (U)}{\lor} s p N_{x}^{θ} (V) = s p N_{x}^{θ} (f^{- 1} (V)),$

and $\begin{array}{l} [f \in α_{3}] \\ = & \underset{x \in A}{\land} \underset{U \in P (Y)}{\land} \min (1 - N_{f (x)}^{β} (U) + \underset{V \in P (X), f (V) \subseteq U}{\lor} N_{f (x)}^{β} (V)) \\ = & \underset{x \in A}{\land} \underset{U \in P (Y)}{\land} \min (1 - N_{f (x)}^{β} (U) + s p N_{x}^{θ} (f^{- 1} (U))) \\ = & [f \in α_{2}] \end{array}$

(4) We prove ⊨f ∈ α ₄ ↔ f ∈ α ₅. First, we show [f ∈ α ₄] ≤ [f ∈ α ₅]. For any B ∈ P (Y), one can deduce that [f ^{- 1} (f $(s p C l_{θ}^{X} (f^{- 1} (B)))) \supseteq s p C l_{θ}^{X} (f$ ^{- 1} (B))] =1, $[{Cl}_{β}^{Y} (f$ (f $- 1 (B))) \subseteq C l_{β}^{Y} (B)] = 1$ , and [f $- 1 (C l_{β}^{Y} (f$ (f ^{- 1} (B)))) ⊆ f $- 1 (C l_{β}^{Y} (B))] = 1.$ From [18, Lemma 1.2(2)], we have $\begin{array}{l} [s p C l_{θ}^{X} (f^{- 1} (B)) \subseteq f^{- 1} (C l_{β}^{Y} (B))] \\ \geq [f & ^{- 1} (f (s p C l_{θ}^{X} (f^{- 1} (B)))) \subseteq (C l_{β}^{Y} (B))] \\ \geq [f & ^{- 1} (f (s p C l_{θ}^{X} (f^{- 1} (B)))) \subseteq (C l_{β}^{Y} (f (f^{- 1} (B))))] \\ \geq [f & (s p C l_{θ}^{X} (f^{- 1} (B))) \subseteq C l_{β}^{Y} (f (f^{- 1} (B)))] . \end{array}$

Therefore, $\begin{array}{l} [f \in α_{5}] \\ = & \underset{B \in P (Y)}{\land} [s p C l_{θ}^{X} (f^{- 1} (B)) \subseteq f^{- 1} (C l_{β}^{Y} (B))] \\ \geq & \underset{B \in P (Y)}{\land} [f (s p C l_{θ}^{X} (f^{- 1} (B))) \subseteq C l_{β}^{Y} (f (f^{- 1} (B)))] \\ \geq & [f (s p C l_{θ}^{X} (A)) \subseteq C l_{β}^{Y} (f (A))] = [f \in α_{4}] \end{array}$

Second, for each A ∈ P (X), there exists a B ∈ P (Y) such that f (A) = B and f ^{- 1} (B) ⊇ A. Hence from [18, Lemma 1.2(1)] we have $\begin{array}{l} \begin{array}{l} [f \in α_{5}] \end{array} \\ \begin{array}{l} = & \underset{A \in P (Y)}{\land} [f (s p C l_{θ}^{X} (A)) \subseteq C l_{β}^{Y} (f (A))] \end{array} \\ \begin{array}{l} \geq & \underset{A \in P (Y)}{\land} [f (s p C l_{θ}^{X} (A)) \subseteq f (f^{- 1} (C l_{β}^{Y} (f (A))))] \end{array} \\ \begin{array}{l} \geq & \underset{A \in P (Y)}{\land} [s p C l_{θ}^{X} (A) \subseteq f^{- 1} (C l_{β}^{Y} (f (A)))] \end{array} \\ \begin{array}{l} \geq & \underset{B \in P (Y), B = f (A)}{\land} [s p C l_{θ}^{X} (f^{- 1} (B)) \subseteq f^{- 1} (C l_{β}^{Y} (A))] \end{array} \\ \begin{array}{l} \geq & \underset{B \in P (Y),}{\land} [s p C l_{θ}^{X} (f^{- 1} (B)) \subseteq f^{- 1} (C l_{β}^{Y} (B))] = [f \in α_{5}] \end{array} \end{array}$

(5) We prove ⊨f ∈ α ₂ ↔ f ∈ α ₅. $\begin{array}{l} \begin{array}{l} [f \in α_{5}] \end{array} \\ \begin{array}{l} = & \underset{B \in P (Y)}{\land} [s p C l_{θ}^{X} (f^{- 1} (B)) \subseteq f^{- 1} (C l_{β}^{Y} (B))] \end{array} \\ \begin{array}{l} = & \underset{B \in P (Y)}{\land} \underset{x \in X}{\land} \end{array} \min (1, 1 - (1 - s p N_{x}^{θ} (X ~ f^{- 1} (B)))) + (1 - (N_{f (x)}^{β} (Y ~ B))) \\ \begin{array}{l} = & \underset{B \in P (Y)}{\land} \underset{x \in X}{\land} \end{array} \min (1, 1 - N_{f (x)}^{β} (Y ~ B) + s p N_{x}^{θ} (f^{- 1} (Y ~ B))) \\ \begin{array}{l} = & \underset{B \in P (Y)}{\land} \underset{x \in X}{\land} \min (1, 1 - N_{f (x)}^{β} (U) + s p N_{x}^{β} (f^{- 1} (U))) \end{array} \\ \begin{array}{l} = & [f \in α_{2}] \end{array} \end{array}$

(6) We prove ⊨f ∈ α ₆ ↔ f ∈ α ₂. $\begin{array}{l} \begin{array}{l} [f \in α_{6}] \end{array} \\ \begin{array}{l} = & \underset{A \subseteq X}{\land} \underset{x \in X}{\land} \min (1, 1 - I n t_{β} (A) (f (x)) + s p I n t_{θ} (f^{- 1} (A)) (x)) \end{array} \\ \begin{array}{l} = & \underset{A \subseteq X}{\land} \underset{x \in X}{\land} \min (1, 1 - N_{f (x)}^{β} (A) + s p N_{x}^{θ} (f^{- 1} (A))) = [f \in α_{2}] . \end{array} \end{array}$

Remark 32. In crisp setting, one have ⊨f ^∈ $I$ _sβ → f ^∈ $I$ _β. But this statement is not true in fuzzifying topological spaces as illustrated by the following example.

Example 33. Let X = {x, y, z}, and $T$ be a fuzzifying topology on X defined in Example 25. Consider the identity function f from (X, $T$ ) onto (X, $U$ ), where $U$ is a fuzzifying topology on X defined as follows: $\begin{array}{l} U (A) = & {\begin{array}{l} 1, A \in {X, \emptyset, {x, y}} \\ 0, otherwise . \end{array} \end{array}$ Then $[I_{s β} (f)] = \frac{1}{8} > 0 = [I_{β} (f)] .$

Remark 34. In crisp setting, one have ⊨f ^∈ $I$ _sβ → f ^∈C _β. But this statement is not true in fuzzifying topological spaces as illustrated by the following example.

Example 35. Let X = {x, y, z}, and $T$ be a fuzzifying topology on X defined in Example 25. Consider the identity function f from (X, $T$ ) onto (X, $U$ ), where $U$ is a fuzzifying topology on X defined as follows: $\begin{array}{l} U (A) = & {\begin{array}{l} 1, A \in {X, \emptyset, {y}} \\ 0, otherwise . \end{array} \end{array}$ Then $[I_{s β} (f)] = \frac{1}{8} > 0 = [C_{β} (f)] .$

Definition 36. Let (X, $T$ ) and (Y, $U$ ) be two fuzzifying topological spaces. f ∈ Y ^X satisfying the following condition is called fuzzifying weak β -irresolute function (write as f ^∈ $I$ _wβ): $ℐ_{w β} (f) = \underset{U \in P (Y)}{\land} \min (1, 1 - U_{s p^{θ}} (U) + ℐ_{β} (f^{- 1} (U))) > 0,$ where $I$ _wβ (f) is called the degree of f being a weak β -irresolute function.

Definition 37. Let (X, $T$ ) and (Y, $U$ ) be two fuzzifying topological spaces, f, and consider the following appointments:

(1) γ ₁ (f): = (∀ $B) (B^{\in} (^{T_{s p θ}) Y} \to (^{f^{- 1} (B)) \in} (^{T_{β}) X}),$ where $(T_{sp θ}^{c})^{Y}$ is the set of all fuzzifying semipre-θ-closed subset of Y and $(T_{β}^{c})^{X}$ is the set of all fuzzifying β -closed subset of X;

(2) γ ₂ (f): = (∀ x) (∀ $U) (U^{\in} sp N_{f (x)}^{θ} \to (f$ $- 1 (U))^{\in} N_{x}^{β}),$ where $sp N_{f (x)}^{θ}$ is the fuzzifying semipre-θ-neighborhood system of f (x) of Y and $N_{x}^{β}$ is the fuzzifying β -neighborhood system of x of X;

(3) γ ₃ (f): = (∀ x) (∀ $U) (U^{\in} sp N_{f (x)}^{θ} \to (\exists V) ((f$ $(V) \subseteq U) \land (V^{\in} N_{x}^{β})));$

(4) γ ₄ (f): = (∀ A) (f $({Cl}_{β}^{X} (A)) \subseteq {spCl}_{θ}^{Y} (f$ (A)));

(5) γ ₅ (f): = (∀ $B) ({Cl}_{β}^{X} (f$ ^{- 1} (B)) ⊆ f $- 1 (s p C l_{θ}^{Y} (B)));$

(6) γ ₆ (f): = (∀ A) (f ^{- 1} (spInt _θ (A)) ⊆ Int _β (f ^{- 1} (A))).

Theorem 38. ⊨f ^∈ $I$ _wβ ↔ f ^∈γ _k, k = 1, …, 6.

Proof. It is similar to the proof of Theorem 31.

Theorem 39. Let (X, $T$ ) and (Y, $U$ ) be two fuzzifying topological space and let f. Then

⊨f ^∈ $I$ _β → f ^∈ $I$ _wβ.

Proof. From Lemma 23(1), we have $U$ _spθ (U) ≤ $U$ _β (U).

Theorem 40. Let (X, $T$ ), (Y, $U$ ) and (Z, ϑ) be three fuzzifying topological spaces, f, and g. Then

(1) ⊨ $I$ _β (f) → ( $I$ _wβ (g) → $I$ _wβ (g o f));

(2) ⊨ $I$ _wβ (g) → ( $I$ _β (f) → $I$ _wβ (g o f)).

Proof. (1) It suffices to show that [ $I$ _β (f)] ≤ [ $I$ _wβ (g) → $I$ _wβ (g ∘ f)]. If [ $I$ _wβ (g)] ≤ [ $I$ _wβ (g ∘ f)], the statement holds. If [ $I$ _wβ (g)] ≥ [ $I$ _wβ (g ∘ f)], then $\begin{array}{l} \begin{array}{l} [ℐ_{w β} (g)] - [ℐ_{w β} (g \circ f)] \end{array} \\ \begin{array}{l} = & \underset{V \in P (Z)}{\land} \min (1, 1 - ϑ_{s p θ} (V) + U_{β} (g^{- 1} (V))) \end{array} \\ \begin{array}{l} - & \underset{V \in P (Z)}{\land} \min (1, 1 - ϑ_{s p θ} (V) + T_{β} ((g \circ f)^{- 1} (V))) \end{array} \\ \begin{array}{l} \leq & \underset{V \in P (Z)}{\land} \min (U_{β} (g^{- 1} (V)) + T_{β} ((g \circ f)^{- 1} (V))) \end{array} \\ \begin{array}{l} = & \underset{V \in P (Z)}{\land} \min (U_{β} (g^{- 1} (V)) + T_{β} (f^{- 1} (f^{- 1} (V)))) \end{array} \\ \begin{array}{l} \leq & \underset{V \in P (Z)}{\land} \min (U_{β} (U) + T_{β} (f^{- 1} (U))) \end{array} \end{array}$

Therefore,

[I _wβ (g) → $I$ _wβ (g ∘ f)] = min(1, 1 - [ $I$ _wβ (g)] + [ $I$ _wβ (g ∘ f)]) min(1, 1 - $U$ _β (U) + $T$ _β (f ^{- 1} (U))) = [ $I$ _β (f)].

$\begin{array}{l} \leq & \underset{U \in P (Y)}{\land} \min (1, 1 - U_{β} (U) + T_{β} (f^{- 1} (U))) = [ℐ_{β} (f)] \end{array}$

(2) $[ℐ_{w β} (g) \to (ℐ_{β} (f) \to ℐ_{w β} (g \circ f))]$

$\begin{array}{l} = [ℐ_{w β} (g) \to \neg (ℐ_{β} (f) \underset{•}{\land} \neg (ℐ_{w β} (g o f)))] \\ = [\neg (ℐ_{w β} (g) \underset{\cdot}{\land} \neg \neg (ℐ_{β} (f) \underset{•}{\land} \neg (ℐ_{w β} (g o f))))] \\ = [\neg (ℐ_{w β} (g) \underset{•}{\land} ℐ_{β} (f) \underset{•}{\land} \neg (ℐ_{w β} (g o f)))] \\ = [\neg (ℐ_{β} (f) \underset{•}{\land} ℐ_{w s p} (g) \underset{•}{\land} \neg (ℐ_{w β} (g o f)))] \\ = [\neg (ℐ_{β} (f) \underset{•}{\land} \neg \neg (ℐ_{w s p} (g) \underset{•}{\land} \neg (ℐ_{w β} (g o f))))] \\ = [ℐ_{β} (f) \to \neg (ℐ_{w β} (g) \underset{•}{\land} \neg (ℐ_{w β} (g o f)))] \\ = [ℐ_{β} (f) \to (ℐ_{w β} (g) \to ℐ_{w β} (g o f))] \end{array}$

5 Conclusions

The present paper investigates topological notions when these are planted into the framework of Ying’s fuzzifying topological spaces (in semantic method of continuous valued-logic). It continue various investigations into fuzzy topology in a legitimate way and extend some fundamental results in general topology to fuzzifying topology. An important virtue of our approach (in which we follow Ying) is that we define topological notions as fuzzy predicates (by formulae of Łukasiewicz fuzzy logic) and prove the validity of fuzzy implications (or equivalences). Unlike the (more wide-spread) style of defining notions in fuzzy mathematics as crisp predicates of fuzzy sets, fuzzy predicates of fuzzy sets provide a more genuine fuzzification; furthermore the theorems in the form of valid fuzzy implications are more general than the corresponding theorems on crisp predicates of fuzzy sets. The main contributions of the present paper are to define fuzzifying semipre-θ-neighborhood system of a point, fuzzifying semipre-θ-closure of a set, fuzzifying semipre-θ-interior of a set, fuzzifying semipre-θ-open sets and fuzzifying semipre-θ-closed sets in the setting fuzzifying topological space. Also, we define the concepts of fuzzifying strong β -irresoluteness and fuzzifying weak β -irresoluteness of fuzzifying topological spaces and obtain some basic properties of such spaces. There are some problems for further study:

(1) One obvious problem is: our results are derived in the Łukasiewicz continuous logic. It is possible to generalize them to more general logic setting, like resituated lattice-valued logic considered in [17, 18].

(2) What is the justification for fuzzifying strong semipre-irresoluteness and fuzzifying weak semipre-irresoluteness in the setting of (2, L) topologies?

(3) Obviously, fuzzifying topological spaces in [19] form a fuzzy category. Perhaps, this will become a motivation for further study of the fuzzy category.

(4) What is the justification for fuzzifying strong β -irresoluteness and fuzzifying weak β -irresoluteness in (M, L)-topologies?

Furthermore, the future possible research of the authors will be to give an application on fuzzifying semipre-θ-topology in solving a decision making problem and medical diagnosis, also the application can be extended to the case ofMV-algebras.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundations of China (11771263, 11641002), the Fundamental Research Funds For the Central Universities (2018CBLY001), and the Fundamental for Graduate students to participate in international academic conference (2018).

Appendix A

Example 35

$U$ (X)	$U$	$U$ _β
∅	1	1
X	1	1
{x}	0	1
{y}	1	1
{z}	0	1
{x, y}	0	1
{x, z}	0	0
{y, z}	0	1

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Strong β -irresoluteness and weak β -irresoluteness based on continuous valued logic

Abstract

Keywords

1 Introduction

1 We take the value lattice as [0, 1], the standard MV-algebra, just for the purpose of readableness. Actually, the results in this paper can be extended to the case of MV-algebras

3 Fuzzifying semipre-θ-closureand fuzzifying semipre-θ-interior

4 Fuzzifying strong β -irresolute and fuzzifying weak β -irresolute functions

5 Conclusions

Footnotes

Acknowledgments

Appendix A

References

1
We take the value lattice as [0, 1], the standard MV-algebra, just for the purpose of readableness. Actually, the results in this paper can be extended to the case of MV-algebras