Fuzzifying bornological linear spaces

Abstract

In this paper, a notion of fuzzifying bornological linear spaces is introduced and the necessary and sufficient condition for fuzzifying bornologies to be compatible with linear structure is discussed. The characterizations of convergence and separation in fuzzifying bornological linear spaces are showed. In particular, some examples with respect to linear fuzzifying bornologies induced by probabilistic normed spaces and fuzzifying topological linear spaces are also provided.

Keywords

Fuzzifying bornological linear spaces fuzzifying bornological convergence separation Product linear fuzzifying bornologies quotient linear fuzzifying bornologies

1 Introduction

It is well known that the theory of topological spaces is an important branch of modern analysis. The main difference between topological and bornological spaces is that the former provide a convenient tool to study continuity, and the latter do the same job for boundedness. Hu [15, 16] first introduced the concept of a bornology and a bornological space to define the concept of boundedness in a topological space. Then most of the research related to the theory of bornological spaces is developed into the context of topological linear spaces (see [14, 25]). Beer [2] extended the research to topological spaces without linear structure, the necessary and sufficient conditions are provided for a set to be the family of bounded sets induced by some admissible metric in [2], and it is also showed that all possible nontrivial metric bornologies arise in this manner if and only if the derived set is compact. After that, the theory of general bornological spaces plays a key role in research of convergence structures on hyperspaces [3 , 18] and optimization theory [5], as well as in study of topologies on function spaces [6 , 19].

It is worthy noting that Abel and Šostak [1] generalized the theory of bornological spaces to the context of fuzzy sets in 2011. They studied bornologies on a complete lattice L and gave the concept of an L-bornology, specially when L = [0, 1]. It was also shown that the category L-Born of L-bornological spaces is a topological construct. Afterwards, Paseka et al. [21] provided the necessary and sufficient condition on the complete lattice for the category L-Born to be topological. Paseka et al. [20] got that for certain complete lattices, the category L-VBorn of L-bornological vector spaces is topological over the category Vec of vector spaces. Moreover, they introduced the category L-VBornSys of L-bornological vector systems, and showed that the category L-VBorn is isomorphic to a full reflective subcategory of L-VBornSys in [20]. In 2016, Šostak and UIjane [26] proposed an alternative approach to fuzzification of the concept of bornology and developed a construction of an L-valued bornology on a set from a family of crisp bornologies on the same set. Meanwhile, they investigated the initial and final L-valued bornologies. After that, UIjane and Šostak [27] extended the L-valued bornologies to a complete completely distributive lattice in 2017.

As is mentioned in [14], bornological linear spaces are one class of those spaces that established the general and precise framework in which the fundamental theorems and techniques of functional analysis hold. It may be expected that the study of fuzzy bornological linear spaces will play an important role in fuzzy functional analysis and fuzzy optimization theory. The motivation of the present paper is to introduce the notion of fuzzifying bornological linear spaces and establish a fundamental framework of fuzzifying bornological linear spaces. Based upon [1 , 27], we intend to introduce a notion of fuzzifying bornological linear spaces and we also aim to establish some relationships between the fuzzifying bornological structure and the topological structure of a linear space. The structure of this paper is as follows. Firstly, in Section 2, we present some necessary notions and fundamental results which are used in the sequel. Secondly, we introduce the concept of a fuzzifying bornological linear space and obtain a necessary and sufficient condition for fuzzifying bornologies to be linear. As examples, we will prove that every Menger probabilistic normed space (Briefly, Menger PN-space) and every fuzzifying topological linear space can be endowed with natural linear fuzzifying bornologies. In particular, the linear fuzzifying bornology generated by probabilistic precompact sets of a Menger PN-space is also considered. In addition, the product and quotient fuzzifying bornological linear spaces are discussed in Section 3. Moreover, in Section 4, we show the characterization of fuzzifying bornological convergence and the separation of fuzzifying bornological linear spaces. In particular, we prove that a fuzzifying bornologically convergent sequence is fuzzifying topologically convergent. The necessary and sufficient condition for fuzzifying bornological linear spaces to be separated is discussed after considering the specific description with respect to fuzzifying bornological closed sets. Meanwhile, we also investigate the separation of product and quotient fuzzifying bornological linear spaces as well as fuzzifying bornological linear subspaces.

2 Preliminaries

In this section, we recall some necessary notions and fundamental results which are used in this paper.

Throughout this paper, X always denotes a universe of discourse. 2^X and $F (X)$ denote the classes of all crisp and fuzzy subsets of X, respectively. The notation 2^(X) is the set of all non-empty finite subsets of X. For any mapping F : X → Y, the notation F^→ : 2^X → 2^Y is defined by F^→ (U) ={ F (x) ∈ Y|x ∈ U } for U ∈ 2^X and F^← : 2^Y → 2^X is defined by F^← (V) ={ x ∈ X|F (x) ∈ V } for V ∈ 2^Y, respectively (see [8, 23]). $K$ represents the field of real or complex numbers, the symbol θ denotes the neutral element of a linear space and * will represent a continuous t-norm (see [17, 24]), and Bal (X) denotes the set of all balanced sets in the linear space X. In addition, the notation →_L denotes the Łukasiewicz implication, i.e. a → _Lb = min {1, 1 - a + b}.

Definition 2.1 (Ying [31]) A fuzzifying topology is a mapping τ : 2^X → [0, 1] such that

(FY1) τ (X) = τ (∅) = 1;

(FY2) τ (U ∩ V) ≥ τ (U) ∧ τ (V) for all U, V ∈ 2^X;

(FY3) $τ (⋃_{j \in J} U_{j}) \geq \underset{j \in J}{⋀} τ (U_{j})$ for every family {U_j|j ∈ J } ⊆ 2^X.F : (X, τ) → (Y, δ) is called continuous with respect to the two fuzzifying topologies τ and δ if δ (V) ≤ τ (F^← (V)) holds for all V ∈ 2^Y.

Definition 2.2 (Ying [31]) Let (X, τ) be a fuzzifying topological space. For any x ∈ X, $N_{x} \in F (2^{X})$ is called a fuzzifying neighborhood system of x which is defined as follows: for any A ∈ 2^X, $A \in N_{x} \overset{Δ}{=} (\exists B \in 2^{X}) ((x \in B \subseteq A) \land (B \in τ))$ .Intuitively, the degree to which A is neighborhood at x isN_x (A) = ⋁ _x∈B⊆Aτ (B).

Definition 2.3 (Qiu [22]) Let (X, τ) be a fuzzifying topological space. Suppose that X is a linear space over $K$ . Then (X, τ) is called a fuzzifying topological linear space if it fulfills the following conditions:For any x, y ∈ X and any V ⊆ X with x + y ∈ V,⊨V ∈ τ → _L ((∃ V_x, V_y ∈ 2^X) (V_x + V_y ⊆ V)∧ (V_x ∈ τ ∧ V_y ∈ τ)) .For each $s \in K$ , any x ∈ X, and any V ∈ 2^X with sx ∈ V,⊨V ∈ τ → _L (∃ δ > 0) (∃ V_x ∈ 2^X) ((V_x ∈ τ) $\land (\forall t \in K) (| t - s | < δ \to_{L} t V_{x} \subseteq V))$ .

Theorem 2.4 (Yan [29]) Let (X, τ) be a fuzzifying topological linear space on $K$ and N_θ (·) be its corresponding fuzzifying neighborhood system of the neutral element. Then it has the following properties:N_θ (X) = 1;

∀U ⊆ X, N_θ (U) > 0 ⇒ θ ∈ U;

∀U, V ⊆ X, N_θ (U ∩ V) = N_θ (U) ∧ N_θ (V);

$\forall W \subseteq X, N_{θ} (W) \leq \underset{U + V \subseteq W}{⋁} N_{θ} (U) \land N_{θ} (V)$ ;

∀U ⊆ X, x ∈ X, N_θ (U) > 0 ⇒ ∃ ɛ > 0 such that kx ∈ U for all |k| < ɛ;

∀U ⊆ X, N_θ (U) > a implies there exists a balanced set V ⊆ U such that N_θ (V) > a.Conversely, let X be a linear space over $K$ and consider a set-valued function N_θ (·) : 2^X → [0, 1] which satisfies the conditions (P1)-(P6). Then there exists a fuzzifying topology τ_N on X such that (X, τ_N) is a fuzzifying topological linear space and N_θ (·) is a fuzzifying neighborhood system of the neutral element with respect to τ_N.

Definition 2.5 (Qiu [22]) Let (X, τ) be a fuzzifying topological linear space. Then the unary fuzzy predicates Bd and $BD \in F (2^{X})$ , called fuzzy boundedness and fuzzy complete boundedness, respectively, are defined as follows:

$Bd (A) \overset{Δ}{=} (\forall V \in 2^{X}) (V \in N_{θ} \to_{L} (\exists λ \in K) (A$ ⊆λV)),

$BD (A) \overset{Δ}{=} (\forall V \in 2^{X}) (V \in N_{θ} \to_{L} V \in ω (A))$

for any A ∈ 2^X, where ω (A) = {V ⊆ X | ∃ z₁, z₂, . . . ,

$z_{n} \in A, s . t . A \subseteq ⋃_{i = 1}^{n} (z_{i} + V)}$ .

Intuitively, the degree to which A is bounded is

$[Bd (A)] = \underset{U \subseteq X}{⋀} {1 - N_{θ} (U) | \forall λ \in K, A ⊈ λ U},$

and the degree to which A is complete bounded is $[BD (A)] = \underset{U \subseteq X}{⋀} {1 - N_{θ} (U) | \forall F \in 2^{(A)}, A ⊈$ F + U} .

Definition 2.6 (Šostak and UIjane [26]) Given a cl-monoid (L, ≤ , ∧ , ∨ , *) (see [7]), an (L, *)-valued bornology on a set X is a mapping $B : 2^{X} \to L$ satisfying the following conditions:

(B1) $\forall x \in X \Rightarrow B ({x}) = 1_{L}$ , where 1_L is the top element of L;

(B2) If A ⊆ B ⊆ X then $B (B) \leq B (A)$ ;

(B3) $\forall A, B \subset X \Rightarrow B (A \cup B) \geq B (A) * B (B)$ .

Then the pair $(X, B)$ is called an (L, *)-valued bornological space and the value $B (A)$ is interpreted as the degree of boundedness of a set A in the space $(X, B)$ .

Let $B (X, L, *)$ stand for the family of all (L, *)-valued bornologies on X, the partial order relation ⪯ on $B (X, L, *)$ by setting for all $B_{1}, B_{2} \in B (X, L, *)$ :

$B_{1} ⪯ B_{2} \Leftrightarrow B_{1} (A) \geq B_{2} (A), \forall A \in 2^{X}$ .

If L = [0, 1], then (L, *)-valued bornology $B$ on X is simply called a fuzzifying bornology. The pair $(X, B)$ is called a fuzzifying bornological space.

Definition 2.7 (Šostak and UIjane [26]) A mapping $f : (X, B_{X}) \to (Y, B_{Y})$ between (L, *)-valued bornological spaces is called bounded if $B_{X} (A) \leq B_{Y} (f (A))$ for every A ∈ 2^X.

Remark 2.8 From Definition 2.7, when L = [0, 1], we can define the degree to which f is bounded as

$[Bd (f)] = \underset{A \subseteq X}{⋀} min {1, 1 - B_{X} (A) + B_{Y} (f (A))}$

for every A ∈ 2^X.

It is easily to check that [Bd (f)] =1 for the case the mapping f is bounded.

Definition 2.9 (Chang et al. [11], Hadžíc and Pap [13]). A Menger PN-space is a triple $(X, F, *)$ , where X is a real linear space, $F$ is a mapping of X into $D$ (Briefly, we shall denote the distributive function $F (x)$ by f_x and $D$ is the set of all distributive functions) satisfying the following conditions:

(1) f_x (t) =1 for all t > 0 if and only if x = θ;

(2) f_x (0) =0;

(3) $f_{kx} (t) = f_{x} (\frac{t}{| k |})$ for all $k \in ℝ$ , k ≠ 0;

(4) f_x+y (t₁ + t₂) ≥ f_x (t₁) * f_y (t₂) for all x, y ∈ X and $t_{1}, t_{2} \in ℝ^{+}$ .

3 Fuzzifying bornological linear spaces

The aim of this section is to introduce the notation of fuzzifying bornological linear spaces. The necessary and sufficient condition on a nonempty set for fuzzifying bornologies to be linear will be discussed. Meanwhile, the product and quotient linear fuzzifying bornologies are considered. Some examples about linear fuzzifying bornologies induced by Menger PN-spaces and fuzzifying topological linear spaces are also provided.

Definition 3.1. Let X be a linear space over $K$ . A fuzzifying bornology $B$ on X is said to be a linear fuzzifying bornology on X, if the following two mappings are bounded:

f : X × X → X, (x, y) ↦ x + y;

$g : K \times X \to X, (k, x) \mapsto kx$ ,

where X × X and $K \times X$ are equipped with the corresponding product fuzzifying bornologies $B \times B$ and $B_{K} \times B$ (here $B_{K}$ is a fuzzifying bornology determined by the crisp bornology on $K$ ) which is defined as

$(B \times B) (A \times B) = \underset{A_{1} \times B_{1} \supseteq A \times B}{⋁} B (A_{1}) \land B (B_{1})$

for all A, B, A₁, B₁ ⊆ X.

We call any pair $(X, B)$ consisting of a linear space and a linear fuzzifying bornology a fuzzifying bornological linear space on X.

Remark 3.2. A usual bornological linear space (see [14]) can be regard as a fuzzifying bornological linear space (with the fuzzifying bornology determined by crisp bornology). Moreover, Paseka et al. gave the definition of L-valued bornological vector spaces in [20]. Here we extend bornological vector space to fuzzifying bornological linear spaces.

In the following, we will present a necessary and sufficient condition on a nonempty set for fuzzifying bornologies to be linear.

Theorem 3.3. Let X be a linear space over $K$ and let $B$ be a fuzzifying bornology on X. Then $B$ is a linear fuzzifying bornology if and only if it satisfies the following conditions: for all A, B ⊆ X(B4) $B (A + B) \geq B (A) \land B (B)$ ;(B5) $B (λ A) \geq B (A)$ , for all $λ \in K$ ;(B6) $B (⋃_{| α | \leq 1} α A) \geq B (A)$ .

Proof. Necessity.

(B4) Since f : X × X → X is bounded, then for all A, B ⊆ X, we have

$B (A + B) = B (f (A \times B)) \geq (B \times B) (A \times B)$

$= \underset{A_{1} \times B_{1} \supseteq A \times B}{⋁} B (A_{1}) \land B (B_{1})$

$\geq B (A) \land B (B)$ .

(B5) Since $g : K \times X \to X$ is bounded, then for all $λ \in K$ and A ⊆ X, we obtain $B (λ A) = B (g ({λ} \times A)) \geq (B_{K} \times B) ({λ} \times A)$

$= \underset{A_{1} \supseteq A}{⋁} B (A_{1}) \geq B (A)$ .

(B6) Let A₂ = {λ||λ|≤1} and for all A ⊆ X, it follows that $B (⋃_{| α | \leq 1} α A) = B (g (A_{2} \times A))$

$\geq \underset{A_{1} \supseteq A}{⋁} B (A_{1}) \geq B (A)$ .

This means that the necessity is proved.

Sufficiency. First, we have $(B \times B) (A) = \underset{B_{1} \times B_{2} \supseteq A}{⋁} B (B_{1}) \land B (B_{2})$

$\leq B (B_{1} + B_{2}) \leq B (f (A))$

for all A ⊆ X × X and B₁, B₂ ⊆ X. It is obvious that f is bounded for all A ⊆ X × X.

Next, we will prove that g is bounded. Let $(B_{K} \times B) (A) = \underset{A \subseteq λ B_{2}, λ \in B_{1}}{⋁} B (B_{2}) > t$ for all $A \subseteq K \times X$ , then there exist $B_{1} \subseteq K$ and B₂ ⊆ X such that A ⊆ B₁ × B₂ and $B (B_{2}) > t$ . Thus,

$B (g (A)) \geq B (g (B_{1} \times B_{2})) = B (⋃_{α \in B_{1}} α B_{2})$ .

It is clear that $B_{K} (B_{1}) = 1$ . Then we get B₁ ⊆ {λ||λ| ≤ m}, where m is a constant in $K$ . It follows that

$B (⋃_{α \in B_{1}} α B_{2}) \geq B (m ⋃_{| α | \leq 1} α B_{2}) > t$ ,

which implies $B (g (A)) \geq (B_{K} \times B) (A)$ . That means g is bounded. This completes the proof.

Example 3.4. (A linear fuzzifying bornology induced by bounded sets of a fuzzifying topological linear space) Let (X, τ) be a fuzzifying topological linear space, then Bd given by Definition 2.5 is a linear fuzzifying bornology.

Proof. The reader can find the proofs of (B1)-(B5) in [30, Proposition 3.3]. We only need to show (B6). It is clear that $\underset{U \subseteq X}{⋀} {1 - N_{θ} (U) | \forall λ \in K, ⋃_{| α | \leq 1} α A ⊈ λ U}$

$\geq \underset{U \subseteq X}{⋀} {1 - N_{θ} (U) | \forall λ \in K, A ⊈ λ U}$ . Then Bd is a linear fuzzifying bornology on X.

Now, we give a special case. For the real line $ℝ$ , we define a fuzzifying topology $τ \in F (2^{ℝ})$ as follows: For any $a, b \in ℝ$ and a < b, $τ ([a, b]) = τ ([a, b)) = τ (a) = \frac{1}{2}$ , $τ ((a, b]) = \frac{3}{4}$ and τ ((a, b)) =1. Then the set $B = {(a, b)} \cup {[a, b)} \cup {(a, b]} \cup ℝ$ defines a base for τ in the sense that for any nonempty set $C \in 2^{ℝ}$ , $a, b \in ℝ$ and a < b, we define $τ (C) = \underset{⋃_{} O_{i} = C, O_{i} \in B}{⋁} \underset{i \in J}{⋀} τ (O_{i})$ and τ (∅) =1, where J is a nonempty index set. From [22, Example 3.1], we know that $(ℝ, τ)$ is a fuzzifying topological vector space. Then we have [Bd ([a, b])] =1 and $[Bd ([a, b))] = \frac{1}{2}$ .

Example 3.5. (A linear fuzzifying bornology induced by bounded sets of a Menger PN-space) Let $(X, F, \land)$ be a Menger PN-space and A ∈ 2^X. Then

$[Bd (A)] = \underset{t > 0}{⋁} \underset{x \in A}{⋀} f_{x} (t)$

(see [28]) is a linear fuzzifying bornology.

Proof. (B1) is clear, we only prove (B2)-(B6). (B2) Let [Bd (A)] < a for A ⊆ B. Then for all t > 0 there exists x ∈ A such that f_x (t) < a. Since x ∈ A ⊆ B, we obtain [Bd (B)] < a easily. Thus, we get [Bd (A)] ≥ [Bd (B)].

(B3) Let [Bd (A)] * [Bd (B)] > a. Then [Bd (A)] ∧ [Bd (B)] ≥ [Bd (A)] * [Bd (B)] > a and there exist t₁, t₂ > 0 such that f_x (t₁) ∧ f_y (t₂) > a for all x ∈ A and y ∈ B. Setting t = t₁ ∨ t₂, it is clear that [Bd (A ∪ B)] ≥ a and [Bd (A ∪ B)] ≥ [Bd (A)] * [Bd (B)].

(B4) By the inequality f_x+y (t₁ + t₂) ≥ f_x (t₁) ∧ f_y (t₂), taking the same method used in (B3), we may easily obtain [Bd (A + B)] ≥ [Bd (A)] ∧ [Bd (B)].

(B5) Let [Bd (A)] > a. Then there exists t > 0 such that f_x (t) > a for all x ∈ A. If λ = 0, it is clear that [Bd (λA)] =1. Suppose that λ ≠ 0, we have $f_{λ x} (t_{1}) = f_{x} (\frac{t_{1}}{| λ |})$ . Setting t₁ = |λ|t, we obtain f_λx (|λ|t) > a for all λx ∈ λA. Hence, [Bd (λA)] > a. Then [Bd (λA)] ≥ [Bd (A)], as desired.

(B6) Let [Bd (A)] > a. Then there exists t > 0 such that f_x (t) > a for all x ∈ A. Thus, we have bx ∈ ⋃ _|α|≤1αA with |b|≤1. If b = 0, we have f_bx (t) = 1 for all t > 0. Suppose that 0 < |b|≤1, we obtain $f_{bx} (t_{1}) = f_{x} (\frac{t_{1}}{| b |}) \geq f_{x} (t_{1})$ . Setting t₁ = t, then it is obvious that [Bd (⋃ _|α|≤1αA)] > a and [Bd (⋃ _|α|≤1αA)] ≥ [Bd (A)].

Thus, we obtain that Bd is a linear fuzzifying bornology.

For a special case,

let $f_{x} (t) = {\begin{matrix} 0 t \leq 0; \\ \frac{1}{2} 0 < t \leq ∥ x ∥; \\ 1 t > ∥ x ∥ . \end{matrix}$

It is easy to see that f_x (t) is a probabilistic norm and [Bd ({x})] =1. If we define A ={ x| ∥ x ∥ > 1 }, then $[Bd (A)] = \frac{1}{2}$ .

Example 3.6. (A linear fuzzifying bornology induced by probabilistic precompact sets of a Menger PN-space) Let $(X, F, \land)$ be a Menger PN-space and A ∈ 2^X. Then

$[BD (A)] = \underset{t > 0}{⋀} \underset{F \in 2^{(A)}}{⋁} \underset{p \in A}{⋀} \underset{q \in F}{⋁} f_{p - q} (t)$

(see [28]) is a linear fuzzifying bornology.

Proof. We need to show $\underset{t > 0}{⋀} \underset{F \in 2^{(A)}}{⋁} \underset{p \in A}{⋀} \underset{q \in F}{⋁} f_{p - q} (t) = \underset{t > 0}{⋀} \underset{F \in 2^{(X)}}{⋁} \underset{p \in A}{⋀} \underset{q \in F}{⋁} f_{p - q} (t)$ at first.

Suppose that $\underset{t > 0}{⋀} \underset{F \in 2^{(X)}}{⋁} \underset{p \in A}{⋀} \underset{q \in F}{⋁} f_{p - q} (t) > 1 - a$ . Then for all t > 0, let $s = \frac{t}{2}$ , there exist F_s ∈ 2^(X) such that for all p ∈ A, ∃q ∈ F with f_p-q (s) > 1 - a. Denote F_s = {x₁, ⋯ , x_n} and N (s, a) = {y | f_y (s) >1 - a}. Then $A \subseteq ⋃_{i = 1}^{n} [x_{i} + N (s, a)]$ . Without loss of generality, we may let A⋂ (x_i + N (s, a)) ¬ = ∅ for all i = 1, 2, ⋯ , n. For any i, there exists z_i ∈ A, y_i ∈ N (s, a) such that z_i = x_i + y_i. Put M = {z₁, z₂, ⋯ , z_n} ∈2^(A). It is easy to verify that $A \subseteq ⋃_{i = 1}^{n} [z_{i} + N (t, a)]$ . It deduces that $⋁_{M \in 2^{(A)}} \underset{p \in A}{⋀} \underset{q \in M}{⋁} f_{p - q} (t) \geq 1 - a$ . By the arbitrariness of t, we have $\underset{t > 0}{⋀} \underset{F \in 2^{(A)}}{⋁} \underset{p \in A}{⋀} \underset{q \in F}{⋁} f_{p - q} (t) \geq 1 - a$ . Hence $\underset{t > 0}{⋀} \underset{F \in 2^{(A)}}{⋁} \underset{p \in A}{⋀} \underset{q \in F}{⋁} f_{p - q} (t)$

$= \underset{t > 0}{⋀} \underset{F \in 2^{(X)}}{⋁} \underset{p \in A}{⋀} \underset{q \in F}{⋁} f_{p - q} (t)$ .

By the definition of [BD (·)], it remains to prove (B2) and (B6) in the following.

(B2) Let [BD (V)] > a. There exists F ∈ 2^(X) such that for all p ∈ V, ∃q ∈ F with f_p-q (t) > a and all t > 0. Since U ⊆ V, it is obvious that $B (U) > a$ and [BD (U)] ≥ [BD (V)].

(B6) Let [BD (U)] >1 - a. Then for all t > 0, put $s = \frac{t}{2}$ , there exists F_s ∈ 2^(U) such that for all p ∈ U, ∃q ∈ F_s with f_p-q (s) > 1 - a. Setting D = {λ||λ|≤1} and it is clear that D is compact. Suppose that $φ : K \to X$ such that φ (λ) = λx for all x ∈ X. It is clear that φ is a continuous linear mapping. Then we get [BD (qD)] =1 and [BD (⋃ _{q∈F_s}qD)] =1. Thus, there exists a finite set F₁ ⊆ DF_s ⊆ DU such that for each y ∈ DF_s, ∃q₁ ∈ F₁ with f_y-q₁ (s) > 1 - a. For any λ with 0 < |λ|≤1 and p ∈ U, it follows that

f_λp-q₁ (t) ≥ f_λp-λq (s) ∧ f_λq-q₁ (s)

$\geq f_{p - q} (\frac{s}{| λ |}) \land (1 - a)$

≥f_p-q (s) ∧ (1 - a)

≥(1 - a) ∧ (1 - a) =1 - a.

This means that [BD (⋃ _0<|λ|≤1αU)] ≥ 1 - a. If λ = 0, it is clear that f_λp-q₁ (t) > 1 - a. So [BD (⋃ _|α|≤1αU)] > 1 - a. That is to say [BD (⋃ _|α|≤1αU)] ≥ [BD (U)], this completes the proof.

Theorem 3.7. Let ${(X_{j}, B_{j})}_{j \in J}$ be a family of fuzzifying bornological linear spaces indexed by a nonempty set J and $X = \prod_{j \in J} X_{j}$ . For every j ∈ J, let f_j : X → X_j be the canonical projection and let f_j be linear. If the mapping $B : 2^{X} \to [0, 1]$ is defined as follows: 1cm $B (A) = \underset{j \in J}{⋀} B_{j} (f_{j} (A)),$ then $B$ is a linear fuzzifying bornology on X. It is called the product linear fuzzifying bornology and denoted by $B = \prod_{j \in J} B_{j}$ . Moreover, $B$ is the initial linear fuzzifying bornology on X.

Proof. First we need to prove that $B$ satisfies the conditions (B1)-(B6).

(B1) Let x_j = f_j (x), we get

$B ({x}) = \underset{j \in J}{⋀} B_{j} (f_{j} ({x})) = \underset{j \in J}{⋀} B_{j} ({x_{j}}) = 1 .$ (B2) A ⊆ B⇒ $\underset{j \in J}{⋀} B_{j} (f_{j} (A)) \geq \underset{j \in J}{⋀} B_{j} (f_{j} (B))$ ⇒ $B (A) \geq B (B)$ .

(B3) $B (A \cup B) = ⋀_{j \in J} B_{j} (f_{j} (A) \cup f_{j} (B))$

$\geq \underset{j \in J}{⋀} B_{j} (f_{j} (A)) * B_{j} (f_{j} (B))$

$\geq B (A) * B (B)$ .

(B4) $B (A + B) = \underset{j \in J}{⋀} B_{j} (f_{j} (A) + f_{j} (B)) \geq \underset{j \in J}{⋀} B_{j} (f_{j} (A)) \land B_{j} (f_{j} (B)) = B (A) \land B (B)$ .

(B5) $B (λ A) = \underset{j \in J}{⋀} B_{j} (λ f_{j} (A)) \geq \underset{j \in J}{⋀} B_{j} (f_{j} (A)) = B (A) .$

(B6) $B (⋃_{| α | \leq 1} α A) = \underset{j \in J}{⋀} B_{j} (⋃_{| α | \leq 1} α f_{j} (A)) \geq \underset{j \in J}{⋀} B_{j} (f_{j} (A)) = B (A)$ .

Hence, $B$ is a linear fuzzifying bornology on X. In the following, we will prove that $B$ is the initial linear fuzzifying bornology on X. First, it needs to prove that f_j is bounded for every j ∈ J. Since f_j (A) = A_j for all A ⊆ X, we get $B_{j} (A_{j}) = B_{j} (f_{j} (A))$ . Then we obtain $\underset{j \in J}{⋀} B_{j} (A_{j}) = \underset{j \in J}{⋀} B_{j} (f_{j} (A))$ , which implies $B (A) \leq B_{j} (f_{j} (A))$ for all A ⊆ X.

Moreover, it remains to prove that $B$ is the weakest linear fuzzifying bornology with each f_j is bounded. Let $\tilde{B}$ is a linear fuzzifying bornology on X with each f_j is bounded, we obtain

$\tilde{B} (A) \leq \underset{j \in J}{⋀} B_{j} (f_{j} (A)) = B (A) .$

This means that $B ⪯ \tilde{B}$ . Therefore the conclusion holds.

Theorem 3.8. Let $(X_{j}, B_{j})$ be defined as Theorem 3.7 and * =∧. Then $\prod ({B_{j}}^{r}) = {(\prod B_{j})}^{r}$ holds with ${B_{j}}^{r} = {A_{j} | B_{j} (A_{j}) \geq r}$ for all r ∈ [0, 1].

Proof. By the discussion in [26], $B_{j}^{r}$ is a crisp bornology on X_j for all j ∈ J and r ∈ [0, 1].

Suppose that $A_{j}, B_{j} \in B_{j}^{r}$ , then $A_{j} + B_{j} \in B_{j}^{r}$ since $B_{j} (A_{j} + B_{j}) \geq B_{j} (A_{j}) \land B_{j} (A_{j}) \geq r$ . Meanwhile, we obtain $λ A_{j} \in B_{j}^{r}$ which follows from $B_{j} (λ A_{j}) \geq B_{j} (A_{j}) \geq r$ . It is clear that $B_{j} (⋃_{| α | \leq 1} α A_{j}) \geq B_{j} (A_{j}) \geq r$ , which implies that $⋃_{| α | \leq 1} α A_{j} \in B_{j}^{r}$ . Therefore, $B_{j}^{r}$ is a linear crisp bornology on X_j for all j ∈ J.

If $A \in {(\prod B_{j})}^{r}$ , then $(\prod B_{j}) (A) \geq r$ . From Theorem 3.7, we get $B_{j} (A_{j}) \geq r$ for all j ∈ J with f_j (A) = A_j, which implies $A_{j} \in {B_{j}}^{r}$ . It is clear that $A \in \prod ({B_{j}}^{r})$ , i.e., $\prod ({B_{j}}^{r}) \supseteq {(\prod B_{j})}^{r}$ . Let $\prod_{j \in J} A_{j} \in \prod ({B_{j}}^{r})$ . Put $A = \prod_{j \in J} A_{j}$ , it is easy to find that f_j (A) = A_j for all j ∈ J. Then $\prod B_{j} (A) = ⋀_{j \in J} B_{j} (f_{j} (A)) = ⋀_{j \in J} B_{j} (A_{j}) \geq r$ , thus $A \in {(\prod B_{j})}^{r}$ . Hence $\prod ({B_{j}}^{r}) \subseteq {(\prod B_{j})}^{r}$ . This completes the proof.

Theorem 3.9. Let $(X, B_{X})$ be a fuzzifying bornological linear space, f : X → Y a linear mapping of X onto a linear space Y. If the mapping $B^{Y} : 2^{Y} \to [0, 1]$ defined as follows: $B^{Y} (C) = \underset{C \subseteq f (A)}{⋁} B_{X} (A)$ . Then $B^{Y}$ is a linear fuzzifying bornology on Y, it is called the quotient linear fuzzifying bornology and denoted by $B^{Y} = B_{X} / f$ . Moreover, $B^{Y}$ is the final linear fuzzifying bornology on Y.

Proof. First we need to prove that $B^{Y}$ satisfies the conditions (B1)-(B6). It is obvious that (B1) holds.

(B2) For all U, V ∈ 2^X with U ⊆ V, if $B^{Y} (V) > t$ , then there exists A ⊆ X such that $B_{X} (A) > t$ with V ⊆ f (A). Moreover, we obtain U ⊆ V ⊆ f (A) and $B^{Y} (U) > t$ . That is $B^{Y} (U) \geq B^{Y} (V)$ .

(B3) Let $B^{Y} (U) * B^{Y} (V) > t$ . Then there exist A, B ⊆ X such that U ⊆ f (A), V ⊆ f (B) and $B_{X} (A) * B_{X} (B) > t$ . Thus, we have U ∪ V ⊆ f (A ∪ B) and $B_{X} (A \cup B) \geq B_{X} (A) * B_{X} (B) > t$ . It is clear that $B^{Y} (U \cup V) > t$ and $B^{Y} (U \cup V) \geq B^{Y} (U) * B^{Y} (V)$ .

(B4) Let $B^{Y} (U) \land B^{Y} (V) > t$ . Then there exist A, B ⊆ X such that U ⊆ f (A), V ⊆ f (B) and $B_{X} (A) \land B_{X} (B) > t$ . Thus U + V ⊆ f (A + B) and $B_{X} (A + B) \geq B_{X} (A) \land B_{X} (B) > t$ . It is obvious that $B^{Y} (U + V) > t$ and $B^{Y} (U \cup V) \geq B^{Y} (U) \land B^{Y} (V)$ .

(B5) Let $B^{Y} (U) > t$ . Then there exists A ⊆ X such that U ⊆ f (A) and $B_{X} (A) > t$ . Thus, λU ⊆ f (λA) and $B_{X} (λ A) \geq B_{X} (A) > t$ . Hence $B^{Y} (λ U) \geq B^{Y} (U) > t$ .

(B6) Let $B^{Y} (U) > t$ . Then there exists A ⊆ X such that U ⊆ f (A) and $B_{X} (A) > t$ . It is clear that (⋃ _|α|≤1αU) ⊆ f (⋃ _|α|≤1αA) and $B_{X} (⋃_{| α | \leq 1} α A) \geq B_{X} (A) > t$ . Hence, we have $B^{Y} (⋃_{| α | \leq 1} α U) \geq B^{Y} (U)$ .

Then it needs to prove that f is bounded. It is clear that $B^{Y} (f (A)) = \underset{f (A) \subseteq f (B)}{⋁} B_{X} (B) \geq B_{X} (A)$ for all A ⊆ X. Thus, f is bounded.

Finally it remains to prove that $B^{Y}$ is the strongest linear fuzzifying bornology with f is bounded. For any linear fuzzifying bornology $\tilde{B}$ on Y with f is bounded, we have $\tilde{B} (f (A)) \geq \underset{f (A) \subseteq f (A)}{⋁} B_{X} (A) = B^{Y} (f (A))$

for all A ⊆ X. So, $\tilde{B} ⪯ B^{Y}$ . Then the conclusion holds.

Theorem 3.10. Let $(X, B_{X})$ , f be defined as Theorem 3.9 and * =∧. Then $(B_{X} / f)^{r} = B_{X}^{r} / f$ .

Proof. From Theorem 3.8, $B_{X}^{r}$ is a linear crisp bornology on X and it is obvious that $(B_{X} / f)^{r} \supseteq B_{X}^{r} / f$ . Suppose that $C \in (B_{X} / f)^{r}$ , then $(B_{X} / f) (C) = \underset{C \subseteq f (A)}{⋁} B_{X} (A) \geq r$ . Which implies that there exists A ⊆ X such that $B_{X} (A) \geq r$ with C ⊆ f (A). That means $f (A) \in B_{X}^{r} / f$ since $A \in B_{X}^{r}$ . Therefore, it is clear that $C \in B_{X}^{r} / f$ . Thus, we have $(B_{X} / f)^{r} \subseteq B_{X}^{r} / f$ . Hence, $(B_{X} / f)^{r} = B_{X}^{r} / f$ . This completes the proof.

Theorem 3.11. Let $(X, B)$ be a fuzzifying bornological linear space and X₁ be a subspace of X. The mapping $B_{1} : 2^{X_{1}} \to [0, 1]$ is defined as follows:

$B_{1} (A) = \underset{A = W \cap X_{1}}{⋁} B (W)$ .

Then $(X_{1}, B_{1})$ is a fuzzifying bornological linear space.

Proof. It is obvious that $B_{1}$ satisfies conditions (B1) and (B2). It remains to prove that $B_{1}$ satisfies (B3)-(B6).

(B3) For every $t < B_{1} (A_{1}) * B_{1} (A_{2})$ , then there exist W₁, W₂ ∈ 2^X with A₁ = W₁ ⋂ X₁, A₂ = W₂ ⋂ X₁ such that $t < B (W_{1}) * B (W_{2})$ . Clearly A₁ ⋃ A₂ = (W₁ ⋃ W₂) ⋂ X₁. Thus

$t < B (W_{1}) * B (W_{2}) \leq B (W_{1} ⋃ W_{2})$

$\leq ⋁_{A_{1} ⋃ A_{2} = W \cap X_{1}} B (W) = B_{1} (A_{1} ⋃ A_{2})$ .

(B4) For each $t < B_{1} (A_{1}) \land B_{1} (A_{2})$ , then exist W₁, W₂ ∈ 2^X with A₁ = W₁ ⋂ X₁, A₂ = W₂ ⋂ X₁ such that $t < B (W_{1})$ and $B (W_{2})$ . It follows that $t < B (W_{1}) \land B (W_{2}) \leq B (W_{1} + W_{2})$ and A₁ + A₂ = (W₁ ⋂ X₁) + (W₂ ⋂ X₁) = (W₁ + W₂) ⋂ X₁. Thus $B_{1} (A_{1}) \land B_{1} (A_{2}) \leq B_{1} (A_{1} + A_{2})$ .

(B5) It is easy to check this condition, here we omit it.

(B6) For any $t < B_{1} (A)$ , there exists W ⊆ X with A = W ⋂ X₁ such that $t < B (W)$ . It follows that $t < B (W) \leq B (⋃_{| α | \leq 1} α W)$ and ⋃_|α|≤1αA = (⋃ _|α|≤1αW) ⋂ X₁. Hence $t < B_{1} (⋃_{| α | \leq 1} α A)$ . Furthermore, $B_{1} (A) \leq B_{1} (⋃_{| α | \leq 1} α A)$ . The proof is completed.

4 Convergence and separation in fuzzifying bornological linear spaces

It is well known that the study of Mackey convergence and separation plays an important role in the the theory of classical bornological linear spaces, such as in the proof of Drop Theorem, Ekeland variation principle and the discussion of duality in bornological linear spaces. In this section, we will extend some basic properties such as convergence and separation to fuzzifying bornological linear spaces. We aim to establish some connections between convergence and separation in fuzzifying bornological linear spaces. Meanwhile, we provide the necessary and sufficient condition for fuzzifying bornological linear spaces to be separated after considering the specific description of fuzzifying bornological closed sets.

Definition 4.1. Let $(X, B)$ be a fuzzifying bornological linear space and let {x_n} be a sequence in X. The degree to which x_n is convergent to x bornologically is $[x_{n} \overset{M}{\to} x] = \underset{\binom{A \in Bal (X)}{λ_{n} \to 0}}{⋁} {B (A) : \forall n \in ℕ, x_{n} - x \in λ_{n} A}$ .

Theorem 4.2. Let $(X, B)$ be a fuzzifying bornological linear space. Then for each sequence {x_n} ⊆ X and x ∈ X, $[x_{n} \overset{M}{\to} x] \land [λ_{n} \to λ] \leq [λ_{n} x_{n} \overset{M}{\to} λ x]$ .

Proof. First, we prove $[λ_{n} \to λ] \leq [λ_{n} x \overset{M}{\to} λ x]$ . It is obvious that the statement holds if [λ_n → λ] =0. Let [λ_n → λ] =1. Setting α_n = λ_n - λ and B = {ax||a|≤1}. Then we obtain $B (B) = B (⋃_{| α | \leq 1} α {x}) \geq B ({x}) = 1$ since α_n → 0, which means that $[λ_{n} x \overset{M}{\to} λ x] = 1$ . Next, we show $[x_{n} \overset{M}{\to} x] \land [λ_{n} \to λ] \leq [λ_{n} x_{n} \overset{M}{\to} λ x]$ , i.e., $[x_{n} \overset{M}{\to} x]^{'} \lor [λ_{n} \to λ]^{'} \geq [λ_{n} x_{n} \overset{M}{\to} λ x]^{'}$ . Let $t > [x_{n} \overset{M}{\to} x]^{'}$ with [λ_n → λ] =1, i.e., $t > \underset{A \in Bal (X), α_{n} \to 0}{⋀} {1 - B (A) : x_{n} - x \in α_{n} A}$ . Then there exist A ∈ Bal (X), α_n → 0 such that x_n - x ∈ α_nA and $B (A) > 1 - t$ . We obtain β_n → 0 such that $B (B) = 1$ and λ_nx - λx ∈ β_nB, which follows from $[λ_{n} x \overset{M}{\to} λ x] = 1$ . Thus, we have λ_nx_n - λx = λ_nx_n - λ_nx + λ_nx - λx

∈β_nB + α_nA ⊆ max {β_n, α_n} (B + A) and $B (B + A) \geq B (A) > 1 - t$ . That means $t > [λ_{n} x_{n} \overset{M}{\to} λ x]^{'}$ , which completes the proof.

Theorem 4.3. Let $(X, B)$ be a fuzzifying bornological linear space. Then for each sequence {x_n} , {y_n} ⊆ X and x, y ∈ X, the following statement holds: $[x_{n} \overset{M}{\to} x] \land [y_{n} \overset{M}{\to} y] \leq [x_{n} + y_{n} \overset{M}{\to} x + y]$ .

Proof. Let $t > [(x_{n} \overset{M}{\to} x)]^{'} \lor [y_{n} \overset{M}{\to} y]^{'}$ . Then there exist A₁, A₂ ∈ Bal (X), α_n, λ_n → 0 such that x_n - x ∈ α_nA₁, y_n - y ∈ λ_nA₂ and $B (A_{1}) \land B (A_{2}) > 1 - t$ . Since $B (A_{1} + A_{2}) \geq B (A_{1}) \land B (A_{2}) > 1 - t$ and x_n + y_n - x - y ∈ α_nA₁ + λ_nA₂ ⊆ max {α_n, λ_n} (A₁ + A₂), we get $t > [x_{n} + y_{n} \overset{M}{\to} x + y]^{'}$ . Furthermore, we obtain $[x_{n} \overset{M}{\to} x] \land [y_{n} \overset{M}{\to} y] \leq [x_{n} + y_{n} \overset{M}{\to} x + y]$ , as desired.

Theorem 4.4. Let $(X, B_{X})$ be a fuzzifying bornological linear space and $f : (X, B_{X}) \to (Y, B_{Y})$ is a linear mapping. Then $[Bd (f)] \leq ⋀_{{x_{n}} \subseteq X} ([x_{n} \overset{M}{\to} x] \to_{L} [f (x_{n}) \overset{M}{\to} f (x)])$ .

Proof. We need to show that $[Bd (f)] \leq 1 - [x_{n} \overset{M}{\to} x] + [f (x_{n}) \overset{M}{\to} f (x)]$ for each sequence {x_n} ⊆ X. It suffices to prove that $t < 1 - [x_{n} \overset{M}{\to} x] + [f (x_{n}) \overset{M}{\to} f (x)]$ for every t < [Bd (f)], Equivalently, we want to prove $[x_{n} \overset{M}{\to} x] \leq 1 - t + [f (x_{n}) \overset{M}{\to} f (x)]$ . For any $r < [x_{n} \overset{M}{\to} x]$ , there exist A ∈ Bal (X) and λ_n → 0 such that $B_{X} (A) > r$ and x_n - x ∈ λ_nA for all $n \in ℕ$ . From the fact f is linear, it follows f (A) ∈ Bal (Y) and f (x_n) - f (x) ∈ λ_nf (A) for all $n \in ℕ$ . Since t < [Bd (f)], it deduces $t < 1 - B_{X} (A) + B_{Y} (f (A))$ . Then $B_{Y} (A) > t + B_{X} (A) - 1 > t + r - 1$ . Thus

$[f (x_{n}) \overset{M}{\to} f (x)] = \underset{\binom{B \in Bal (Y)}{α_{n} \to 0}}{⋁} {B_{Y} (B) : f (x_{n}) - f (x) \in α_{n} B} \geq B_{Y} (A) > t + r - 1$ .

So $1 - t + [f (x_{n}) \overset{M}{\to} f (x)] > r$ . By the arbitrariness of r, we have $1 - t + [f (x_{n}) \overset{M}{\to} f (x)] \geq [x_{n} \overset{M}{\to} x]$ . This completes the proof.

Corollary 4.5. Let $(X, B_{X})$ be a fuzzifying bornological linear space and $f : (X, B_{X}) \to (Y, B_{Y})$ is a bounded linear mapping. Then, $[x_{n} \overset{M}{\to} x] \leq [f (x_{n}) \overset{M}{\to} f (x)]$ for each sequence {x_n} ⊆ X and x ∈ X.

Definition 4.6 (Ying [31]) Let (X, τ) be a fuzzifying topological space. Then for any x ∈ X and any S ∈ D (X), we define $S Z x \overset{Δ}{=} (\forall V \in 2^{X}) ((V \in N_{x}) \to_{L} S ⊑ V)$ .

Where the notation S ⊑ V means S almost in V, that is, there is n₀ ∈ D such that S (n) ∈ V for all n ∈ D with n₀ ≺ n and D (X) = {S|S : D → X, (D, ≺) textupisadirectset}. Intuitively, the value of S converges to x, that is [S Z x] is $[S Z x] = \underset{S ⋢ V}{⋀} (1 - N_{x} (V))$ .

Theorem 4.7. Let (X, τ) be a fuzzifying topological linear space and $(X, B)$ be a fuzzifying bornological linear space. If we define $B (A) = [Bd (A)]$ for all A ⊆ X, then $[x_{n} \overset{M}{\to} x] \leq [x_{n} \overset{τ}{\to} x]$ .

Proof. Let $[x_{n} \overset{M}{\to} x] = \underset{A \in Bal (X), λ_{n} \to 0}{⋁} {B (A) : x_{n} - x \in λ_{n} A} > t$ . Then, there exist A ∈ Bal (X), λ_n → 0 such that x_n - x ∈ λ_nA and $B (A) > t$ . Hence, $B (A) = [Bd (A)]$

$= \underset{U \subseteq X}{⋀} {1 - N_{θ} (U) | \forall λ \in K, A ⊈ λ U} > t$ . We obtain 1 - N_θ (U) > t if A ⊈ λU for all U ⊆ X and $λ \in K$ .

On the other hand, from the property (P6) in Theorem 2.4, it is easy to check that $[x_{n} \overset{τ}{\to} x] = ⋀_{x_{n} - x ⋢ V} (1 - N_{θ} (V)) = ⋀_{x_{n} - x ⋢ V, V \in Bal (X)} (1 - N_{θ} (V))$ . Then for any balanced set V with x_n - xnotsqsubseteqV, it follows that AnotsubseteqλV for all λ. Otherwise, if there is λ₀ such that A ⊆ λ₀V, it deduces that x_n - x ∈ λ_nA ⊆ λ_n λ₀V for all $n \in ℕ$ . Since λ_n λ₀ → 0, there is $N_{0} \in ℕ$ such that |λ_n λ₀|<1 for all n ≥ N₀. Then x_n - x ∈ λ_n λ₀V ⊆ V for all n ≥ N₀, i.e., x_n - x ⊑ V, this contradicts with the hypothesis x_n - xnotsqsubseteqV. Thus 1 - N_θ (V) > t. So $[x_{n} \overset{τ}{\to} x] \geq t$ , which completes the proof.

Definition 4.8. Let $(X, B)$ be a fuzzifying bornological linear space. Then a unary predicate $T \in F (Σ)$ called separation is defined as follows: $(X, B) \in T \overset{Δ}{=} ((\forall M \in Svec (X)) ((M \in B) \to_{L} (M = {θ})),$

where the notation Svec (X) denotes all linear subspaces of X.

Moreover, the degree to which $(X, B)$ is separated is 8mm $[T (X, B)] = \underset{\binom{M \neq {θ}}{M \in Svec (X)}}{⋀} {1 - B (M)} .$

Theorem 4.9. Let $(X, B)$ be a fuzzifying bornological linear space. Then for each sequence {x_n} ⊆ X and x, y ∈ X, the following statement holds: $([x_{n} \overset{M}{\to} x] \land [x_{n} \overset{M}{\to} y]) ⊓ [T (X, B)] \leq [x = y]$ ,

where the notation ⊓ is the Łukasiewicz t-norm, i.e.,a ⊓ b = max {a + b - 1, 0}.

Proof. We only need to show that $[x_{n} \overset{M}{\to} x] \land [y_{n} \overset{M}{\to} y] \leq {[T (X, B)]}^{'}$ if x ≠ y. Let $[x_{n} \overset{M}{\to} x] \land [x_{n} \overset{M}{\to} y] > t$ . Then, there exist A, B ∈ Bal (X), λ_n, α_n → 0 such that x_n - x ∈ λ_nA, x_n - y ∈ α_nB and $B (A) \land B (B) > t$ . Hence, y - x = x_n - x - x_n + y ∈ λ_nA + α_nB ⊆ max {λ_n, α_n} (A + B). For any $λ \in K,$ from the fact max {λ_n, α_n} →0, it follows that λ (y - x) ∈ λ max {λ_n, α_n} (A + B) ⊆ (A + B) for some n. Thus $B (span {y - x}) \geq B (A + B) > t$ . Hence, $[T (X, B)]^{'} = \underset{M \neq {θ}}{⋁} B (M) > t .$ The proof is completed.

Theorem 4.10. Let $(X, B)$ be a fuzzifying bornological linear space. Then for each sequence {x_n} ⊆ X and x, y ∈ X, the following statement holds: $⋀_{{x_{n}} \subseteq X} [(x_{n} \overset{M}{\to} x) \land (x_{n} \overset{M}{\to} y) \to_{L} (x = y)]$

$\leq [T (X, B)]$ .

Proof. We need to show that $[T (X, B)] \geq \underset{{x_{n}} \subseteq X}{⋀} {1 - [x_{n} \overset{M}{\to} x] \land [y_{n} \overset{M}{\to} y] + [x = y]}$ . Let $[T (X, B)] = \underset{M \neq {θ}}{⋀} {1 - B (M)} < t$ . Then, there exists M ≠ {θ} such that $1 - B (M) < t$ . Setting A = span {x} for all x ∈ M, we get $B (A) > 1 - t$ . It is clear that nx ∈ A for every $n \in ℕ$ . Let x_n = x. Then we obtain $[x_{n} \overset{M}{\to} x] = 1$ and $[x_{n} \overset{M}{\to} θ] = \underset{B \in Bal (X), λ_{n} \to 0}{⋁} {B_{X} (B) : x_{n} \in λ_{n} B} > 1 - t$ . Hence, $\underset{{x_{n}} \subseteq X}{⋀} {1 - [x_{n} \overset{M}{\to} x] \land [y_{n} \overset{M}{\to} y] + [x = y]} < t$ . That is $[T (X, B)] \geq \underset{{x_{n}} \subseteq X}{⋀} {1 - [x_{n} \overset{M}{\to} x] \land [y_{n} \overset{M}{\to} y] + [x = y]}$ , which completes the proof.

Theorem 4.11. Let $(X, B)$ be a fuzzifying bornological linear space. Then, $[T (X, B)] = λ$ (λ ∈ [0, 1]) if and only if for each r > 1 - λ, $(X, B^{r})$ is separated and $(X, B^{r})$ is not separated for each r < 1 - λ with $B^{r} = {A \subseteq X | B (A) \geq r}$ .

Proof. Necessity. It is clear that $B^{r} = {A \subseteq X | B (A) \geq r}$ is a linear crisp bornology. Since

$[T (X, B)] = \underset{\binom{M \neq {θ}}{M \in Svec (X)}}{⋀} {1 - B (M)} = λ$ .

Then for r > 1 - λ, we obtain $B (M) < r$ for all M ≠ {θ} , M ∈ Svec (X). Hence, we have M = {θ} for every linear space $M \in B^{r}$ . This means that $(X, B^{r})$ is separated. For every r ∈ (0, 1 - λ), there is a linear space M ≠ {θ} such that $B (M) > r$ . Obviously $M \in B^{r}$ . It follows that $(X, B^{r})$ is not separated.

Sufficiency. Since for each r > 1 - λ, $(X, B^{r})$ is separated, it follows that $[T (X, B)] \leq λ$ . If $[T (X, B)] < λ$ , it deduces a contradiction with the hypothesis $(X, B^{r})$ is not separated for each r < 1 - λ. Hence $[T (X, B)] = λ$ .

Definition 4.12. Let $(X, B)$ be a fuzzifying bornological linear space. Then a unary predicate $BC \in F (Σ)$ called bornologically closed is defined as follows:

$A \in BC \overset{Δ}{=} (\forall {x_{n}} \subseteq A) ((x_{n} \overset{M}{\to} x) \to_{L} (x \in A)) .$

Moreover, the degree to which A is bornologically closed is $[BC (A)] = \underset{\binom{{x_{n}} \subseteq A}{x \notin A}}{⋀} \underset{\binom{B \in Bal (X)}{λ_{n} \to 0}}{⋀} {1 - B (B) : \forall n \in ℕ, x_{n} - x \in λ_{n} B} .$

Theorem 4.13. Let $(X, B)$ be a fuzzifying bornological linear space. Then $BC ({θ}) = [T (X, B)]$ .

Proof. From Theorem 4.10, it is clear $BC ({θ}) \leq T [(X, B)]$ . For each t > BC ({θ}), there exist B ∈ Bal (X) , x ¬ = θ and λ_n → 0 such that $1 - B (B) < t$ and x ∈ λ_nB. Thus, we obtain $T [(X, B)] < t$ which follows from B ≠ {θ} and $1 - B (B) < t$ . Therefore, the inequality $BC ({θ}) \geq T [(X, B)]$ holds, as desired. That means $BC ({θ}) = T [(X, B)]$ , which completes the proof.

Corollary 4.14. Let $(X, B)$ be a fuzzifying bornological linear space, M a subspace of X and $B^{X / M}$ be the quotient linear fuzzifying bornology on X/M. Then $BC (M) = T [(X / M, B^{X / M})]$ .

Proof. It is clear M is the zero element in X/M, From Theorem 4.13, $T [(X / M, B^{X / M})] = BC ({\hat{θ}}) = BC (M)$ .

Theorem 4.15. With the notion given in Theorem 3.7, the following statements hold: (a) $(⋀_{j \in J} [T (X_{j}, B_{j})]) \land [(\forall x \in X) (x \neq θ) \to_{L} (\exists j \in J) (f_{j} (x) \neq θ_{j})] \leq [T (X, B)]$ ; (b) $(\underset{j \in J}{⋀} [T (X_{j}, B_{j})]) \land [T (X, B)] \leq [(\forall x \in X) (x \neq θ) \to_{L} (\exists j \in J) (f_{j} (x) \neq θ_{j})]$ .

Proof. (a) We only prove the case that [(∀ x ∈ X) (x ≠ θ) → _L (∃ j ∈ J) (f_j (x) ≠ θ_j)] =1. Let $[T (X, B)] < t$ . Then, there exists M ≠ {θ} with M ∈ Svec (X) such that $B (M) > 1 - t$ . From the boundedness of f_j for all j ∈ J, we obtain $B_{j} (f_{j} (M)) \geq B (M) > 1 - t$ . Suppose that there exists j ∈ J such that f_j (x) ≠ θ_j for any x ≠ θ. It is clear that f_j (M) ≠ {θ_j} , f_j (M) ∈ Svec (X_j). Then it follows that $\underset{j \in J}{⋀} [T (X_{j}, B_{j})] < t$ , which completes the proof of (a). (b) It is obvious that the statement holds if [(∀ x ∈ X) (x ≠ θ) → _L (∃ j ∈ J) (f_j (x) ≠ θ_j)] =1. Suppose that [(∀ x ∈ X) (x ≠ θ) → _L (∃ j ∈ J) (f_j (x) ≠ θ_j)] =0 and $[T (X, B)] > t$ , then for all M ≠ {θ} , M ∈ Svec (X), we have $1 - B (M) > t$ . Thus, we obtain $1 - t > B (M) = \underset{j \in J}{⋀} B_{j} (M_{j})$ with M_j = f_j (M). Since [(∀ x ∈ X) (x ≠ θ) → _L (∃ j ∈ J) (f_j (x) ≠ θ_j)] =0, there exists x ≠ θ such that f_j (x) = θ_j for all j ∈ J. Put M_j = {θ_j}, then $B_{j} (M_{j}) = 1$ . It deduces a contradiction. Thus $[T (X, B)] \leq t$ . By the arbitrariness of t, we get $[T (X, B)] = 0$ , which completes the proof.

Theorem 4.16. Let $(X, B)$ be a fuzzifying bornological linear space and X₁ be a subspace of X. Then $[T (X, B)] \leq [T (X_{1}, B_{1})]$ , where $B_{1}$ is defined as Theorem 3.11.

Proof. Let $[T (X_{1}, B_{1})] < t$ . Then there exists M ≠ {θ} with M ∈ Svec (X₁) such that $B_{1} (M) = \underset{M = W \cap X_{1}}{⋁} B (W) > 1 - t$ . Here we let W = M, thus we obtain W ≠ {θ} and $B (W) > 1 - t$ . Hence we get $T [(X, B)] < t$ , which completes the proof.

Corollary 4.17. Let $(X_{i}, B_{i})$ (i = 1, 2) be fuzzifying bornological linear spaces. Then $[T (X_{1}, B_{1})] \land [T (X_{2}, B_{2})] \leq [T (X_{1} \cap X_{2}, B)]$ , where $B (A) = \underset{A = (W_{1} \cap W_{2}) \cap (X_{1} \cap X_{2})}{⋁} B_{1} (W_{1}) \lor B_{2} (W_{2})$ .

5 Conclusions and future work

In the present paper, we have built a fundamental framework of fuzzifying bornological linear spaces by using many-valued logic. In this framework, a concept of fuzzifying bornological linear spaces is introduced and the necessary and sufficient condition for fuzzifying bornologies to be compatible with linear structure is discussed. Also, we obtained characterizations of convergence and separation in fuzzifying bornological linear spaces. Some examples with respect to linear fuzzifying bornologies induced by probabilistic normed spaces and fuzzifying topological linear spaces are provided.

A direction worthy of future work is to establish (L, M)-fuzzy bornological linear spaces, corresponding to the M-bornologies on L-fuzzy sets mentioned in [27]; also, generalizing further some concepts defined in this paper such as fuzzy bornological convergence and fuzzy bornological closed is of interest.

Footnotes

Acknowledgments

The authors are grateful to the referees and Editorial Board for their valuable comments and helpful suggestions in modifying this paper. We also acknowledge the support of National Natural Science Foundation of China under Grant No.: 11571006, No.:12071225 and Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

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