Fuzzifying bornological linear spaces

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 →  _L b = 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 ∈ J U j ) ≥ ⋀ j ∈ 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 ∈ F ( 2 X ) is called a fuzzifying neighborhood system of x which is defined as follows: for any A ∈ 2 ^X , A ∈ N x = Δ ( ∃ B ∈ 2 X ) ( ( x ∈ B ⊆ A ) ∧ ( B ∈ 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 ∈ K , any x ∈ X, and any V ∈ 2 ^X with sx ∈ V,⊨V ∈ τ →  _L  (∃ δ > 0) (∃ V _x  ∈ 2 ^X ) ((V _x  ∈ τ) ∧ ( ∀ t ∈ K ) ( | t - s | < δ → L t V x ⊆ 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);

∀ W ⊆ X , N θ ( W ) ≤ ⋁ U + V ⊆ W N θ ( U ) ∧ 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 ∈ F ( 2 X ) , called fuzzy boundedness and fuzzy complete boundedness, respectively, are defined as follows:

Bd ( A ) = Δ ( ∀ V ∈ 2 X ) ( V ∈ N θ → L ( ∃ λ ∈ K ) ( A ⊆λV)),

BD ( A ) = Δ ( ∀ V ∈ 2 X ) ( V ∈ N θ → L V ∈ ω ( A ) )

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

z n ∈ A , s . t . A ⊆ ⋃ i = 1 n ( z i + V ) } .

Intuitively, the degree to which A is bounded is

[ Bd ( A ) ] = ⋀ U ⊆ X { 1 - N θ ( U ) | ∀ λ ∈ K , A ⊈ λ U } ,

and the degree to which A is complete bounded is [ BD ( A ) ] = ⋀ U ⊆ X { 1 - N θ ( U ) | ∀ F ∈ 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 → L satisfying the following conditions:

(B1) ∀ x ∈ X ⇒ B ( { x } ) = 1 L , where 1 _L is the top element of L;

(B2) If A ⊆ B ⊆ X then B ( B ) ≤ B ( A ) ;

(B3) ∀ A , B ⊂ X ⇒ B ( A ∪ B ) ≥ 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 ∈ B ( X , L , * ) :

B 1 ⪯ B 2 ⇔ B 1 ( A ) ≥ B 2 ( A ) , ∀ A ∈ 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 ) → ( Y , B Y ) between (L, *)-valued bornological spaces is called bounded if B X ( A ) ≤ 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 ) ] = ⋀ A ⊆ 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 ( t | k | ) for all k ∈ 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 ∈ 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 × X → X , ( k , x ) ↦ kx ,

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

( B × B ) ( A × B ) = ⋁ A 1 × B 1 ⊇ A × B B ( A 1 ) ∧ 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 ) ≥ B ( A ) ∧ B ( B ) ;(B5) B ( λ A ) ≥ B ( A ) , for all λ ∈ K ;(B6) B ( ⋃ | α | ≤ 1 α A ) ≥ 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 × B ) ) ≥ ( B × B ) ( A × B )

= ⋁ A 1 × B 1 ⊇ A × B B ( A 1 ) ∧ B ( B 1 )

≥ B ( A ) ∧ B ( B ) .

(B5) Since g : K × X → X is bounded, then for all λ ∈ K and A ⊆ X, we obtain B ( λ A ) = B ( g ( { λ } × A ) ) ≥ ( B K × B ) ( { λ } × A )

= ⋁ A 1 ⊇ A B ( A 1 ) ≥ B ( A ) .

(B6) Let A₂ = {λ||λ|≤1} and for all A ⊆ X, it follows that B ( ⋃ | α | ≤ 1 α A ) = B ( g ( A 2 × A ) )

≥ ⋁ A 1 ⊇ A B ( A 1 ) ≥ B ( A ) .

This means that the necessity is proved.

Sufficiency. First, we have ( B × B ) ( A ) = ⋁ B 1 × B 2 ⊇ A B ( B 1 ) ∧ B ( B 2 )

≤ B ( B 1 + B 2 ) ≤ 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 × B ) ( A ) = ⋁ A ⊆ λ B 2 , λ ∈ B 1 B ( B 2 ) > t for all A ⊆ K × X , then there exist B 1 ⊆ K and B₂ ⊆ X such that A ⊆ B₁ × B₂ and B ( B 2 ) > t . Thus,

B ( g ( A ) ) ≥ B ( g ( B 1 × B 2 ) ) = B ( ⋃ α ∈ 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 ∈ B 1 α B 2 ) ≥ B ( m ⋃ | α | ≤ 1 α B 2 ) > t ,

which implies B ( g ( A ) ) ≥ ( B K × 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 ⋀ U ⊆ X { 1 - N θ ( U ) | ∀ λ ∈ K , ⋃ | α | ≤ 1 α A ⊈ λ U }

≥ ⋀ U ⊆ X { 1 - N θ ( U ) | ∀ λ ∈ 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 ∈ F ( 2 ℝ ) as follows: For any a , b ∈ ℝ and a < b, τ ( [ a , b ] ) = τ ( [ a , b ) ) = τ ( a ) = 1 2 , τ ( ( a , b ] ) = 3 4 and τ ((a, b)) =1. Then the set B = { ( a , b ) } ∪ { [ a , b ) } ∪ { ( a , b ] } ∪ ℝ defines a base for τ in the sense that for any nonempty set C ∈ 2 ℝ , a , b ∈ ℝ and a < b, we define τ ( C ) = ⋁ ⋃ O i = C , O i ∈ B ⋀ i ∈ 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 ) ) ] = 1 2 .

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

[ Bd ( A ) ] = ⋁ t > 0 ⋀ x ∈ 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 ( 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 ( t 1 | b | ) ≥ 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 ) = { 0 t ≤ 0 ; 1 2 0 < t ≤ ∥ x ∥ ; 1 t > ∥ x ∥ .

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 ) ] = 1 2 .

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

[ BD ( A ) ] = ⋀ t > 0 ⋁ F ∈ 2 ( A ) ⋀ p ∈ A ⋁ q ∈ F f p - q ( t )

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

Proof. We need to show ⋀ t > 0 ⋁ F ∈ 2 ( A ) ⋀ p ∈ A ⋁ q ∈ F f p - q ( t ) = ⋀ t > 0 ⋁ F ∈ 2 ( X ) ⋀ p ∈ A ⋁ q ∈ F f p - q ( t ) at first.

Suppose that ⋀ t > 0 ⋁ F ∈ 2 ( X ) ⋀ p ∈ A ⋁ q ∈ F f p - q ( t ) > 1 - a . Then for all t > 0, let s = 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 ⊆ ⋃ 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 ⊆ ⋃ i = 1 n [ z i + N ( t , a ) ] . It deduces that ⋁ M ∈ 2 ( A ) ⋀ p ∈ A ⋁ q ∈ M f p - q ( t ) ≥ 1 - a . By the arbitrariness of t, we have ⋀ t > 0 ⋁ F ∈ 2 ( A ) ⋀ p ∈ A ⋁ q ∈ F f p - q ( t ) ≥ 1 - a . Hence ⋀ t > 0 ⋁ F ∈ 2 ( A ) ⋀ p ∈ A ⋁ q ∈ F f p - q ( t )

= ⋀ t > 0 ⋁ F ∈ 2 ( X ) ⋀ p ∈ A ⋁ q ∈ 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 = 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 → 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)

≥ f p - q ( s | λ | ) ∧ ( 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 ∈ J be a family of fuzzifying bornological linear spaces indexed by a nonempty set J and X = ∏ j ∈ 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 → [ 0 , 1 ] is defined as follows: 1cm B ( A ) = ⋀ j ∈ 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 = ∏ j ∈ 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 } ) = ⋀ j ∈ J B j ( f j ( { x } ) ) = ⋀ j ∈ J B j ( { x j } ) = 1 . (B2) A ⊆ B⇒ ⋀ j ∈ J B j ( f j ( A ) ) ≥ ⋀ j ∈ J B j ( f j ( B ) ) ⇒ B ( A ) ≥ B ( B ) .

(B3) B ( A ∪ B ) = ⋀ j ∈ J B j ( f j ( A ) ∪ f j ( B ) )

≥ ⋀ j ∈ J B j ( f j ( A ) ) * B j ( f j ( B ) )

≥ B ( A ) * B ( B ) .

(B4) B ( A + B ) = ⋀ j ∈ J B j ( f j ( A ) + f j ( B ) ) ≥ ⋀ j ∈ J B j ( f j ( A ) ) ∧ B j ( f j ( B ) ) = B ( A ) ∧ B ( B ) .

(B5) B ( λ A ) = ⋀ j ∈ J B j ( λ f j ( A ) ) ≥ ⋀ j ∈ J B j ( f j ( A ) ) = B ( A ) .

(B6) B ( ⋃ | α | ≤ 1 α A ) = ⋀ j ∈ J B j ( ⋃ | α | ≤ 1 α f j ( A ) ) ≥ ⋀ j ∈ 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 ⋀ j ∈ J B j ( A j ) = ⋀ j ∈ J B j ( f j ( A ) ) , which implies B ( A ) ≤ 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 B ˜ is a linear fuzzifying bornology on X with each f _j is bounded, we obtain

B ˜ ( A ) ≤ ⋀ j ∈ J B j ( f j ( A ) ) = B ( A ) .

This means that B ⪯ B ˜ . Therefore the conclusion holds.

Theorem 3.8. Let ( X j , B j ) be defined as Theorem 3.7 and * =∧. Then ∏ ( B j r ) = ( ∏ B j ) r holds with B j r = { A j | B j ( A j ) ≥ 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 ∈ B j r , then A j + B j ∈ B j r since B j ( A j + B j ) ≥ B j ( A j ) ∧ B j ( A j ) ≥ r . Meanwhile, we obtain λ A j ∈ B j r which follows from B j ( λ A j ) ≥ B j ( A j ) ≥ r . It is clear that B j ( ⋃ | α | ≤ 1 α A j ) ≥ B j ( A j ) ≥ r , which implies that ⋃ | α | ≤ 1 α A j ∈ B j r . Therefore, B j r is a linear crisp bornology on X _j for all j ∈ J.

If A ∈ ( ∏ B j ) r , then ( ∏ B j ) ( A ) ≥ r . From Theorem 3.7, we get B j ( A j ) ≥ r for all j ∈ J with f _j  (A) = A _j , which implies A j ∈ B j r . It is clear that A ∈ ∏ ( B j r ) , i.e., ∏ ( B j r ) ⊇ ( ∏ B j ) r . Let ∏ j ∈ J A j ∈ ∏ ( B j r ) . Put A = ∏ j ∈ J A j , it is easy to find that f _j  (A) = A _j for all j ∈ J. Then ∏ B j ( A ) = ⋀ j ∈ J B j ( f j ( A ) ) = ⋀ j ∈ J B j ( A j ) ≥ r , thus A ∈ ( ∏ B j ) r . Hence ∏ ( B j r ) ⊆ ( ∏ 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 → [ 0 , 1 ] defined as follows: B Y ( C ) = ⋁ C ⊆ 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 ) ≥ 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 ∪ B ) ≥ B X ( A ) * B X ( B ) > t . It is clear that B Y ( U ∪ V ) > t and B Y ( U ∪ V ) ≥ B Y ( U ) * B Y ( V ) .

(B4) 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 U + V ⊆ f (A + B) and B X ( A + B ) ≥ B X ( A ) ∧ B X ( B ) > t . It is obvious that B Y ( U + V ) > t and B Y ( U ∪ V ) ≥ B Y ( U ) ∧ 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 ) ≥ B X ( A ) > t . Hence B Y ( λ U ) ≥ 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 | ≤ 1 α A ) ≥ B X ( A ) > t . Hence, we have B Y ( ⋃ | α | ≤ 1 α U ) ≥ B Y ( U ) .

Then it needs to prove that f is bounded. It is clear that B Y ( f ( A ) ) = ⋁ f ( A ) ⊆ f ( B ) B X ( B ) ≥ 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 B ˜ on Y with f is bounded, we have B ˜ ( f ( A ) ) ≥ ⋁ f ( A ) ⊆ f ( A ) B X ( A ) = B Y ( f ( A ) )

for all A ⊆ X. So, 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 ⊇ B X r / f . Suppose that C ∈ ( B X / f ) r , then ( B X / f ) ( C ) = ⋁ C ⊆ f ( A ) B X ( A ) ≥ r . Which implies that there exists A ⊆ X such that B X ( A ) ≥ r with C ⊆ f (A). That means f ( A ) ∈ B X r / f since A ∈ B X r . Therefore, it is clear that C ∈ B X r / f . Thus, we have ( B X / f ) r ⊆ 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 → [ 0 , 1 ] is defined as follows:

B 1 ( A ) = ⋁ A = W ∩ 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 ) ≤ B ( W 1 ⋃ W 2 )

≤ ⋁ A 1 ⋃ A 2 = W ∩ X 1 B ( W ) = B 1 ( A 1 ⋃ A 2 ) .

(B4) For each t < B 1 ( A 1 ) ∧ 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 ) ∧ B ( W 2 ) ≤ B ( W 1 + W 2 ) and A₁ + A₂ = (W₁ ⋂ X₁) + (W₂ ⋂ X₁) = (W₁ + W₂) ⋂ X₁. Thus B 1 ( A 1 ) ∧ B 1 ( A 2 ) ≤ 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 ) ≤ B ( ⋃ | α | ≤ 1 α W ) and ⋃_|α|≤1αA = (⋃ _|α|≤1αW) ⋂ X₁. Hence t < B 1 ( ⋃ | α | ≤ 1 α A ) . Furthermore, B 1 ( A ) ≤ B 1 ( ⋃ | α | ≤ 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 → M x ] = ⋁ A ∈ Bal ( X ) λ n → 0 { B ( A ) : ∀ n ∈ ℕ , x n - x ∈ 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 → M x ] ∧ [ λ n → λ ] ≤ [ λ n x n → M λ x ] .

Proof. First, we prove [ λ n → λ ] ≤ [ λ n x → M 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 | ≤ 1 α { x } ) ≥ B ( { x } ) = 1 since α _n  → 0, which means that [ λ n x → M λ x ] = 1 . Next, we show [ x n → M x ] ∧ [ λ n → λ ] ≤ [ λ n x n → M λ x ] , i.e., [ x n → M x ] ′ ∨ [ λ n → λ ] ′ ≥ [ λ n x n → M λ x ] ′ . Let t > [ x n → M x ] ′ with [λ _n  → λ] =1, i.e., t > ⋀ A ∈ Bal ( X ) , α n → 0 { 1 - B ( A ) : x n - x ∈ α n A } . Then there exist A ∈ Bal (X), α _n  → 0 such that x _n  - x ∈ α _n A and B ( A ) > 1 - t . We obtain β _n  → 0 such that B ( B ) = 1 and λ _n x - λx ∈ β _n B, which follows from [ λ n x → M λ x ] = 1 . Thus, we have λ _n x _n  - λx = λ _n x _n  - λ _n x + λ _n x - λx

∈β _n B + α _n A ⊆ max {β _n , α _n } (B + A) and B ( B + A ) ≥ B ( A ) > 1 - t . That means t > [ λ n x n → M 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 → M x ] ∧ [ y n → M y ] ≤ [ x n + y n → M x + y ] .

Proof. Let t > [ ( x n → M x ) ] ′ ∨ [ y n → M y ] ′ . Then there exist A₁, A₂ ∈ Bal (X), α _n , λ _n  → 0 such that x _n  - x ∈ α _n A₁, y _n  - y ∈ λ _n A₂ and B ( A 1 ) ∧ B ( A 2 ) > 1 - t . Since B ( A 1 + A 2 ) ≥ B ( A 1 ) ∧ B ( A 2 ) > 1 - t and x _n  + y _n  - x - y ∈ α _n A₁ + λ _n A₂ ⊆ max {α _n , λ _n } (A₁ + A₂), we get t > [ x n + y n → M x + y ] ′ . Furthermore, we obtain [ x n → M x ] ∧ [ y n → M y ] ≤ [ x n + y n → M x + y ] , as desired.

Theorem 4.4. Let ( X , B X ) be a fuzzifying bornological linear space and f : ( X , B X ) → ( Y , B Y ) is a linear mapping. Then [ Bd ( f ) ] ≤ ⋀ { x n } ⊆ X ( [ x n → M x ] → L [ f ( x n ) → M f ( x ) ] ) .

Proof. We need to show that [ Bd ( f ) ] ≤ 1 - [ x n → M x ] + [ f ( x n ) → M f ( x ) ] for each sequence {x _n } ⊆ X. It suffices to prove that t < 1 - [ x n → M x ] + [ f ( x n ) → M f ( x ) ] for every t < [Bd (f)], Equivalently, we want to prove [ x n → M x ] ≤ 1 - t + [ f ( x n ) → M f ( x ) ] . For any r < [ x n → M x ] , there exist A ∈ Bal (X) and λ _n  → 0 such that B X ( A ) > r and x _n  - x ∈ λ _n A for all n ∈ ℕ . From the fact f is linear, it follows f (A) ∈ Bal (Y) and f (x _n ) - f (x) ∈ λ _n f (A) for all n ∈ 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 ) → M f ( x ) ] = ⋁ B ∈ Bal ( Y ) α n → 0 { B Y ( B ) : f ( x n ) - f ( x ) ∈ α n B } ≥ B Y ( A ) > t + r - 1 .

So 1 - t + [ f ( x n ) → M f ( x ) ] > r . By the arbitrariness of r, we have 1 - t + [ f ( x n ) → M f ( x ) ] ≥ [ x n → M x ] . This completes the proof.

Corollary 4.5. Let ( X , B X ) be a fuzzifying bornological linear space and f : ( X , B X ) → ( Y , B Y ) is a bounded linear mapping. Then, [ x n → M x ] ≤ [ f ( x n ) → M 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 = Δ ( ∀ V ∈ 2 X ) ( ( V ∈ N x ) → 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 ] = ⋀ 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 → M x ] ≤ [ x n → τ x ] .

Proof. Let [ x n → M x ] = ⋁ A ∈ Bal ( X ) , λ n → 0 { B ( A ) : x n - x ∈ λ n A } > t . Then, there exist A ∈ Bal (X), λ _n  → 0 such that x _n  - x ∈ λ _n A and B ( A ) > t . Hence, B ( A ) = [ Bd ( A ) ]

= ⋀ U ⊆ X { 1 - N θ ( U ) | ∀ λ ∈ K , A ⊈ λ U } > t . We obtain 1 - N _θ  (U) > t if A ⊈ λU for all U ⊆ X and λ ∈ K .

On the other hand, from the property (P6) in Theorem 2.4, it is easy to check that [ x n → τ x ] = ⋀ x n - x ⋢ V ( 1 - N θ ( V ) ) = ⋀ x n - x ⋢ V , V ∈ 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 _n A ⊆ λ _n λ₀V for all n ∈ ℕ . Since λ _n λ₀ → 0, there is N 0 ∈ 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 → τ x ] ≥ t , which completes the proof.

Definition 4.8. Let ( X , B ) be a fuzzifying bornological linear space. Then a unary predicate T ∈ F ( Σ ) called separation is defined as follows: ( X , B ) ∈ T = Δ ( ( ∀ M ∈ Svec ( X ) ) ( ( M ∈ B ) → 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 ) ] = ⋀ M ≠ { θ } M ∈ 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 → M x ] ∧ [ x n → M y ] ) ⊓ [ T ( X , B ) ] ≤ [ 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 → M x ] ∧ [ y n → M y ] ≤ [ T ( X , B ) ] ′ if x ≠ y. Let [ x n → M x ] ∧ [ x n → M y ] > t . Then, there exist A, B ∈ Bal (X), λ _n , α _n  → 0 such that x _n  - x ∈ λ _n A, x _n  - y ∈ α _n B and B ( A ) ∧ B ( B ) > t . Hence, y - x = x _n  - x - x _n  + y ∈ λ _n A + α _n B ⊆ max {λ _n , α _n } (A + B). For any λ ∈ 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 } ) ≥ B ( A + B ) > t . Hence, [ T ( X , B ) ] ′ = ⋁ M ≠ { 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 } ⊆ X [ ( x n → M x ) ∧ ( x n → M y ) → L ( x = y ) ]

≤ [ T ( X , B ) ] .

Proof. We need to show that [ T ( X , B ) ] ≥ ⋀ { x n } ⊆ X { 1 - [ x n → M x ] ∧ [ y n → M y ] + [ x = y ] } . Let [ T ( X , B ) ] = ⋀ M ≠ { 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 ∈ ℕ . Let x _n  = x. Then we obtain [ x n → M x ] = 1 and [ x n → M θ ] = ⋁ B ∈ Bal ( X ) , λ n → 0 { B X ( B ) : x n ∈ λ n B } > 1 - t . Hence, ⋀ { x n } ⊆ X { 1 - [ x n → M x ] ∧ [ y n → M y ] + [ x = y ] } < t . That is [ T ( X , B ) ] ≥ ⋀ { x n } ⊆ X { 1 - [ x n → M x ] ∧ [ y n → M y ] + [ x = y ] } , which completes the proof.