M -fuzzifying cotopological spaces and M -fuzzifying convex spaces as M -fuzzifying closure spaces

1 Introduction

By a closure system, we usually mean a subset of the power set 2 ^X which satisfies certain conditions [5]. With the development of fuzzy set theory, closure systems are generalized to different kinds of fuzzy closure systems. Biacino and Gerla [3, 4] defined a kind of fuzzy closure systems by extending 2 ^X to I ^X . Moreover, Kim [16] proved that the lattice of fuzzy closure systems is isomorphic to the lattice of fuzzy closure operators for the respective notions defined in [4]. Later, Bělohlávek [2] outlined a general theory of fuzzy closure operators and fuzzy closure systems (L _K -closure systems), and showed the existence of a one-to-one correspondence between his fuzzy closure operators and fuzzy closure systems.

However, the above-mentioned fuzzy closure systems are crisp families of fuzzy subsets on a universe set X. Following the idea of [11, 35], Fang [7] proposed the concept of L-fuzzy closure systems. Unluckily, L-fuzzy closure operators in [7] are not equivalent to L-fuzzy closure systems. Later, Luo and Fang [19] introduced the concepts of fuzzifying closure systems and fuzzifying closure operators as a generalization of Birkhoff’s closure operators. Moreover, a one-to-one correspondence between the notions was established. Actually, the relations between closure systems and closure operators are investigated extensively [6, 40].

In this paper, we will focus on another important aspect of closure systems. As we all know, a closure system is not only a generalization of a cotopology [14] by relaxing the condition that it is closed under finite unions, but also a generalization of a convex structure [36] by relaxing the condition that it is closed under directed unions. In a categorical sense, the category of cotopological spaces and the category of convex spaces are both subcategories of the category of closure spaces. In [21, 38], convex structures were generalized to the fuzzy case and were widely studied. In the framework of fuzzy topology, categorical relations between different kinds of spatial structures are investigated extensively [12, 22–24]. Motivated by this, we will provide a categorical approach to investigate the relations between different kinds of spatial structures. In this paper, we will first generalize fuzzifying closure systems in [19] to M-fuzzifying closure systems. Then we attempt to show that M-fuzzifying cotopologies [10, 39] and M-fuzzifying convex structures [33] can be included in the framework of M-fuzzifying closure systems [19] from a categorical aspect.

2 Preliminaries

A complete lattice M is said to be completely distributive if it satisfies the completely distributive law (CD law, in short), i.e., ⋁ i ∈ I ⋀ j ∈ J i a i , j = ⋀ f ∈ ∏ i ∈ I J i ⋁ i ∈ I a i , f ( i ) or ⋀ i ∈ I ⋁ j ∈ J i a i , j = ⋁ f ∈ ∏ i ∈ I J i ⋀ i ∈ I a i , f ( i ) holds for all X _i  = {a_i,j ∣ j ∈ J _i } ⊆2 ^M  (i ∈ I).

Throughout this paper, let M denote a completely distributive lattice. The smallest element and the largest element in M are denoted by ⊥ and ⊤, respectively. For a, b ∈ M, we say that a is wedge below b in M, in symbols a ≺ b, if for every subset D ⊆ L, ⋁D ⩾ b implies d ⩾ a for some d ∈ D. A complete lattice M is completely distributive if and only if b = ⋁ {a ∈ M ∣ a ≺ b} for each b ∈ M. For a nonempty set X, 2 ^X denotes the powerset of X. Let { A j } j ∈ J ⊆ dir 2 X denote that {A _j } _j∈J is a directed subset of 2 ^X , which means for each B, C ∈ {A _j } _j∈J, there exists D ∈ {A _j } _j∈J such that B ⊆ D and C ⊆ D. Let ⋃ dir j ∈ J A j denote the directed union of { A j } j ∈ J ⊆ dir 2 X .

Let X, Y be two nonempty sets and φ : X ⟶ Y be a mapping. Define φ^→ : 2 ^X  ⟶ 2 ^Y and φ^← : 2 ^Y  ⟶ 2 ^X as follows: ∀ A ∈ 2 X , φ → ( A ) = { φ ( x ) ∣ x ∈ A } ; ∀ B ∈ 2 Y , φ ← ( B ) = { x ∣ φ ( x ) ∈ B } .

Definition 2.1. [19] An M-fuzzifying closure system on X is a mapping S : 2 X ⟶ M which satisfies:

S ( ∅ ) = S ( X ) = ⊤ ;

For an M-fuzzifying closure system S on X, the pair ( X , S ) is called an M-fuzzifying closure space.

A mapping φ : ( X , S X ) ⟶ ( Y , S Y ) between M-fuzzifying closure spaces is called continuous if it satisfies ∀ B ∈ 2 Y , S X ( φ ← ( B ) ) ⩾ S Y ( B ) .

It is easy to check that all M-fuzzifying closure spaces as objects and all continuous mappings as morphisms form a category, denoted by M-FYS.

As we all know, topologies and cotopologies are dual and equivalent. In the situation of M-fuzzifying topology theory, M-fuzzifying topologies are also equivalent to M-fuzzifying cotopologies. In the sequel, we will adopt the concept of M-fuzzifying cotopologies.

Definition 2.2. [10, 39] An M-fuzzifying cotopology on X is a mapping τ : 2 ^X  ⟶ M which satisfies:

τ (∅) = τ (X) = ⊤;

For an M-fuzzifying cotopology τ on X, the pair (X, τ) is called an M-fuzzifying cotopological space.

A mapping φ : (X, τ _X ) ⟶ (Y, τ _Y ) between M-fuzzifying cotopological spaces is called continuous if it satisfies ∀ B ∈ 2 Y , τ X ( φ ← ( B ) ) ⩾ τ Y ( B ) .

It is easy to check that all M-fuzzifying cotopological spaces as objects and all continuous mappings as morphisms form a category, denoted by M-FYT.

Definition 2.3. [33] An M-fuzzifying convex structure on X is a mapping C : 2 X ⟶ M which satisfies:

C ( ∅ ) = C ( X ) = ⊤ ;

For an M-fuzzifying convex structure C on X, the pair ( X , C ) is called an M-fuzzifying convex space.

A mapping φ : ( X , C X ) ⟶ ( Y , C Y ) between M-fuzzifying convex spaces is called convexity preserving (CP, in short) if it satisfies ∀ B ∈ 2 Y , C X ( φ ← ( B ) ) ⩾ C Y ( B ) .

It is easy to check that all M-fuzzifying convex spaces as objects and all convexity preserving mappings as morphisms form a category, denoted by M-FYC.

Actually, it is easy to see that M-FYT and M-FYC are both full subcategories of M-FYS.

Definition 2.4. [1, 27] A category C is called a topological category over Set with respect to the usual forgetful functor from C to Set if it satisfies (TC1)–(TC3).

(TC1) Existence of final structures: For any set X, any class J, and family ((X _j , ξ _j )) _j∈J of C-object and any family (f _j  : X _j  ⟶ X) _j∈J of mappings, there exists a unique C-structure ξ on X which is final with respect to the sink (f _j  : (X _j , ξ _j ) ⟶ X)) _j∈J, this means that for a C-object (Y, η), a mapping g : (X, ξ) ⟶ (Y, η) is a C-morphism iff for all j ∈ J, g ∘ f _j  : (X _j , ξ _j ) ⟶ (Y, η) is a C-morphism.

(TC2) Fibre-smallness: For any set X, the C-fibre of X, i.e., the class of all C-structures on X, which we denote by C(X), is a set.

(TC3) Terminal separator property: For any set X with cardinality at most one, there exists exactly one C-object with underlying set X (i.e. there exists exactly one C-structure on X).

For more notions related to category theory we refer to [1] and [27].

3 M-FYT as M-FYS

An M-fuzzifying closure system is just an M-fuzzifying cotopology by relaxing (MFYT3). In this sense, M-fuzzifying closure systems are generalizations of M-fuzzifying cotopologies. In this section, we will investigate the relations between M-fuzzifying closure systems and M-fuzzifying cotopologies from a categorical aspect.

Lemma 3.1.Let ( X , S ) be an M-fuzzifying closure space and define S ★ : 2 X ⟶ M as follows: ∀ A ∈ 2 X , S ★ ( A ) = ⋁ ∪ j ∈ F A j = A ∧ j ∈ F S ( A j ) , where F is finite. Then S ★ satisfies (MFYT3). Moreover, S ★ ( A ) ⩾ S ( A ) ( ∀ A ∈ 2 X ) .

Proof. Take any A, B ∈ 2 ^X . Then S ★ ( A ) ∧ S ★ ( B ) = ⋁ ∪ i ∈ F 1 A i = A ∧ i ∈ F 1 S ( A i ) ∧ ⋁ ∪ j ∈ F 2 B j = B ∧ j ∈ F 2 S ( B j ) = ⋁ ∪ i ∈ F 1 A i = A ⋁ ∪ j ∈ F 2 B j = B ∧ i ∈ F 1 ∧ j ∈ F 2 ( S ( A i ) ∧ S ( B j ) ) ⩽ ⋁ ( ∪ i ∈ F 1 A i ) ∪ ( ∪ j ∈ F 2 B j ) = A ∪ B ∧ i ∈ F 1 ∧ j ∈ F 2 ( S ( A i ) ∧ S ( B j ) ) ⩽ ⋁ ∪ k ∈ F C k = A ∪ B ∧ k ∈ F S ( C k ) = S ★ ( A ∪ B ) .

Moreover, S ★ ( A ) ⩾ S ( A ) ( ∀ A ∈ 2 X ) holds obviously. □

Based on Lemma 3.1, we can construct anM-fuzzifying cotopology from an M-fuzzifying closure system.

Theorem 3.2. Let ( X , S ) be an M-fuzzifying closure space and define S ¯ : 2 X ⟶ M as follows: ∀ A ∈ 2 X , S ¯ ( A ) = ⋁ ⋂ j ∈ J A j = A ⋀ j ∈ J S ★ ( A j ) .

Then S ¯ is an M-fuzzifying cotopology on X. Moreover, S ¯ ( A ) ⩾ S ( A ) ( ∀ A ∈ 2 X ) .

Proof. By the definition of S ¯ , it follows that S ¯ ( A ) ⩾ S ★ ( A ) ⩾ S ( A ) . Then it suffices to verify that S ¯ satisfies (MFYT1)–(MFYT3). In fact,

(MFYT1) Since S ¯ ( A ) ⩾ S ( A ) , by (MFYS1), we have S ¯ ( ∅ ) = S ¯ ( X ) = ⊤ .

(MFYT2) For{A _j  ∣ j ∈ J} ⊆2 ^X , we have ⋀ j ∈ J S ¯ ( A j ) = ⋀ j ∈ J ⋁ ⋂ k ∈ K j A k , j = A j ⋀ k ∈ K j S ★ ( A k , j ) = ⋁ F ∈ ∏ j ∈ J A j ⋀ j ∈ J ⋀ B ∈ F ( j ) S ★ ( B ) ( by CD law , where A j = { { A k , j } k ∈ K j ∣ ⋂ k ∈ K j A k , j = A j } ) ⩽ ⋁ ⋂ m ∈ M B m = ⋂ j ∈ J A j ⋀ m ∈ M S ★ ( B m ) = S ¯ ( ⋂ j ∈ J A j ) .

(MFYT3) For each A, B ∈ 2 ^X , by Lemma 3.1, it follows that S ¯ ( A ) ∧ S ¯ ( B ) = ⋁ ⋂ j ∈ J A j = A ⋀ j ∈ J S ★ ( A j ) ∧ ⋁ ⋂ k ∈ K B k = B ⋀ k ∈ K S ★ ( B k ) = ⋁ ⋂ j ∈ J A j = A ⋁ ⋂ k ∈ K B k = B ⋀ j ∈ J ⋀ k ∈ K S ★ ( A j ) ∧ S ★ ( B k ) ⩽ ⋁ ⋂ j ∈ J A j = A ⋁ ⋂ k ∈ K B k = B ⋀ j ∈ J ⋀ k ∈ K S ★ ( A j ∪ B k ) ⩽ ⋁ ⋃ j ∈ J ⋃ k ∈ K A j ∪ B k = A ∪ B ⋀ j ∈ J ⋀ k ∈ K S ★ ( A j ∪ B k ) ⩽ ⋁ ⋂ m ∈ M C m = A ∪ B ⋀ m ∈ M S ★ ( C m ) = S ¯ ( A ∪ B ) .

This shows that S ¯ is an M-fuzzifying cotopology on X. □

Lemma 3.3. Let (Y, τ _Y ) be an M-fuzzifying cotopological space and φ : Y ⟶ X be a mapping. Define τ _X  : 2 ^X  ⟶ M as follows: ∀ A ∈ 2 X , τ X ( A ) = τ Y ( φ ← ( A ) ) .

Then τ _X is an M-fuzzifying cotopology on X.

Proof. It suffices to show that τ _X satisfies (MFYT1)–(MFYT3).

(MFYT1) τ _X  (∅) = τ _Y  (φ^← (∅)) = τ _Y  (∅) = ⊤ and τ _X  (X) = τ _Y  (φ^← (X)) = τ _Y  (Y) =⊤.

(MFYT2) For {A _j  ∣ j ∈ J} ⊆2 ^X , it follows that τ X ( ⋂ j ∈ J A j ) = τ Y ( φ ← ( ⋂ j ∈ J A j ) ) = τ Y ( ⋂ j ∈ J φ ← ( A j ) ) ⩾ ⋀ j ∈ J τ Y ( φ ← ( A j ) ) = ⋀ j ∈ J τ X ( A j ) .

(MFYT3) Take any A, B ∈ 2 ^X . Then τ X ( A ∪ B ) = τ Y ( φ ← ( A ∪ B ) ) = τ Y ( φ ← ( A ) ∪ φ ← ( B ) ) ⩾ τ Y ( φ ← ( A ) ) ∧ τ Y ( φ ← ( B ) ) = τ X ( A ) ∧ τ X ( B ) . □

Theorem 3.4. M-MFYT is a bicoreflective subcategory of M-MFYS.

Proof. Let ( X , S ) be an M-fuzzifying closure space. By Theorem 3.2, we obtain S ¯ is an M-fuzzifying cotopology on X. Next we claim that id X : ( X , S ¯ ) ⟶ ( X , S ) is the M-MFYT-bicoreflector.

For this it suffices to show:

id X : ( X , S ¯ ) ⟶ ( X , S ) is continuous.

For (1), by Theorem 3.2, we know for each A ∈ 2 ^X , it follows that S ¯ ( A ) ⩾ S ( A ) . This implies that id X : ( X , S ¯ ) ⟶ ( X , S ) is continuous.

For (2), for each M-fuzzifying cotopological space (Y, τ _Y ) and each mapping φ : Y ⟶ X, define τ _X as in Lemma 3.3. Then (X, τ _X ) is an M-fuzzifying cotopological space. Since φ : ( Y , τ Y ) ⟶ ( X , S ) is continuous, we have ∀ A ∈ 2 X , τ X ( A ) = τ Y ( φ ← ( A ) ) ⩾ S ( A ) .

Further, by the definition of S ¯ , we obtain S ¯ ( A ) = ⋁ ⋂ j ∈ J A j = A ⋀ j ∈ J S ★ ( A j ) ⩽ ⋁ ⋂ j ∈ J A j = A ⋀ j ∈ J ⋁ ∪ k ∈ F B j , k = A j ∧ k ∈ F S ( B j , k ) ⩽ ⋁ ⋂ j ∈ J A j = A ⋀ j ∈ J ⋁ ∪ k ∈ F B j , k = A j ∧ k ∈ F τ X ( B j , k ) ⩽ ⋁ ⋂ j ∈ J A j = A ⋀ j ∈ J ⋁ ∪ k ∈ F B j , k = A j τ X ( ∪ k ∈ F B j , k ) = ⋁ ⋂ j ∈ J A j = A ⋀ j ∈ J τ X ( A j ) ⩽ ⋁ ⋂ j ∈ J A j = A τ X ( ⋂ j ∈ J A j ) = τ X ( A ) = τ Y ( φ ← ( A ) ) .

This proves that φ : ( Y , τ Y ) ⟶ ( X , S ¯ ) is continuous. □

4 M-FYC as M-FYS

In this section, we will investigate the relations between M-fuzzifying closure systems and M-fuzzifying convex structures. By Definitions 2.1 and 2.3, it is easy to see that an M-fuzzifying closure system is exactly an M-fuzzifying convex structure by relaxing (MFYC3). In the sequel, we will investigate their categorical relations.

Theorem 4.1. Let ( X , S ) be an M-fuzzifying closure space and define S * : 2 X ⟶ M as follows: ∀ A ∈ 2 X , S * ( A ) = ⋁ ⋃ dir k ∈ K A k = A ⋀ k ∈ K S ( A k ) .

Then S * is an M-fuzzifying convex structure on X. Moreover, S * ( A ) ⩾ S ( A ) ( ∀ A ∈ 2 X ) .

Proof. It suffices to show that S * satisfies (MFYC1)–(MFYC3). In fact,

(MFYC1) Since S satisfies (MFYS1), it follows that S * ( ∅ ) = S * ( X ) = ⊤ .

(MFYC2) Take {A _i  ∣ i ∈ I} ⊆2 ^X . Then we prove S * ( ⋂ i ∈ I A i ) ⩾ ⋀ i ∈ I S * ( A i ) , i.e., ⋁ ⋃ dir k ∈ K B k = ⋂ i ∈ I A i ⋀ k ∈ K S ( B k ) ⩾ ⋀ i ∈ I ⋁ ⋃ dir j ∈ J i G i , j = A i ⋀ j ∈ J i S ( G i , j ) .

Take any a ∈ M such that a ≺ ⋀ i ∈ I ⋁ ⋃ dir j ∈ J i G i , j = A i ⋀ j ∈ J i S ( G i , j ) .

Then for each i ∈ I, there exists a directed set {G_i,j ∣ j ∈ J _i } ⊆2 ^X such that ⋃ dir j ∈ J i G i , j = A i and for each j ∈ J _i , S ( G i , j ) ⩾ a . Thus, ⋂ i ∈ I A i = ⋂ i ∈ I ⋃ j ∈ J i G i , j = ⋃ f ∈ ∏ i ∈ I J i ⋂ i ∈ I G i , f ( i ) and S ( G i , f ( i ) ) ⩾ a . Since {G_i,j ∣ j ∈ J _i } is directed, it is trivial to check that { ⋂ i ∈ I G i , f ( i ) ∣ f ∈ ∏ i ∈ I J i } is directed. This implies that ⋁ ⋃ dir k ∈ K B k = ⋂ i ∈ I A i ⋀ k ∈ K S ( B k ) ⩾ ⋀ f ∈ ∏ i ∈ I J i S ( ⋂ i ∈ I G i , f ( i ) ) ⩾ ⋀ f ∈ ∏ i ∈ I J i ⋀ i ∈ I S ( G i , f ( i ) ) ⩾ a .

By the arbitrariness of a, we obtain S * ( ⋂ i ∈ I A i ) ⩾ ⋀ i ∈ I S * ( A i ) , as desired.

(MFYC3) Take { A j ∣ j ∈ J } ⊆ dir 2 X . Then we prove that S * ( ⋃ dir j ∈ J A j ) ⩾ ⋀ j ∈ J S * ( A j ) . That is to say, ⋁ ⋃ dir k ∈ K B k = ⋃ dir j ∈ J A j ⋀ k ∈ K S ( B k ) ⩾ ⋀ j ∈ J ⋁ ⋃ dir i ∈ I j G j , i = A j ⋀ i ∈ I j S ( G j , i ) .

Take any a ∈ M such that a ≺ ⋀ j ∈ J ⋁ ⋃ dir i ∈ I j G j , i = A j ⋀ i ∈ I j S ( G j , i ) .

Then for each j ∈ J, there exists a directed set {G_j,i ∣ i ∈ I _j } such that ⋃ dir i ∈ I j G j , i = A j and for each i ∈ I _j , S ( G j , i ) ⩾ a . Put C = ⋃ dir j ∈ J A j . Then C = ⋃ dir j ∈ J ⋃ dir i ∈ I j G j , i . Now define a mapping σ : 2 fin C ⟶ 2 X as follows: ∀ F ∈ 2 fin C , σ ( F ) = ⋂ { G j , i ∣ F ⊆ G j , i } , where 2 fin C denotes the set of all finite subsets of C. Then it is easy to check that C = ⋃ F ∈ 2 fin C F = ⋃ F ∈ 2 fin C σ ( F ) = ⋃ F ∈ 2 fin C ⋂ F ⊆ G j , i G j , i .

By the definition of σ, we know σ is order preserving. Since { F ∣ F ∈ 2 fin C } is directed, we know { σ ( F ) ∣ F ∈ 2 fin C } is directed. Thus, it follows that S ( σ ( F ) ) = S ( ⋂ F ⊆ G j , i G j , i ) ⩾ ⋀ F ⊆ G j , i S ( G j , i ) ⩾ a .

This implies that ⋁ ⋃ dir k ∈ K B k = ⋃ dir j ∈ J A j ⋀ k ∈ K S ( B k ) ⩾ ⋀ F ⊆ G j , i S ( G j , i ) ⩾ a .

By the arbitrariness of a, we obtain ⋁ ⋃ dir k ∈ K B k = ⋃ dir j ∈ J A j ⋀ k ∈ K S ( B k ) ⩾ ⋀ j ∈ J ⋁ ⋃ dir i ∈ I j G j , i = A j ⋀ i ∈ I j S ( G j , i ) , as desired. □

Lemma 4.2. Let ( Y , C Y ) be an M-fuzzifying convex space and φ : Y ⟶ X be a mapping. Define C X : 2 X ⟶ M as follows: ∀ A ∈ 2 X , C X ( A ) = C Y ( φ ← ( A ) ) .

Then C X is an M-fuzzifying convex structure on X.

Proof. It suffices to show that τ _X satisfies (MFYC1)–(MFYC3).

(MFYC1) C X ( ∅ ) = C Y ( φ ← ( ∅ ) ) = C Y ( ∅ ) = ⊤ and C X ( X ) = C Y ( φ ← ( X ) ) = C Y ( Y ) = ⊤ .

(MFYC2) For {A _i  ∣ i ∈ I} ⊆2 ^X , it follows that C X ( ⋂ i ∈ I A i ) = C Y ( φ ← ( ⋂ i ∈ I A i ) ) = C Y ( ⋂ i ∈ I φ ← ( A i ) ) ⩾ ⋀ i ∈ I C Y ( φ ← ( A i ) ) = ⋀ i ∈ I C X ( A i ) .

(MFYC3) For { A j ∣ j ∈ J } ⊆ dir 2 X , it follows that C X ( ⋃ dir j ∈ J A j ) = C Y ( φ ← ( ⋃ dir j ∈ J A j ) ) = C Y ( ⋃ dir j ∈ J φ ← ( A j ) ) ⩾ ⋀ j ∈ J C Y ( φ ← ( A j ) ) = ⋀ j ∈ J C X ( A j ) .

This shows that C X is an M-fuzzifying convex structure on X. □

Theorem 4.3. M-MFYC is a bicoreflective subcategory of M-MFYS.

Proof. Let ( X , S ) be an M-fuzzifying closure space. By Theorem 4.1, we obtain S * is an M-fuzzifying convex structure on X. Next we claim that id X : ( X , S * ) ⟶ ( X , S ) is the M-MFYC-bicoreflector.

For this it suffices to show:

id X : ( X , S * ) ⟶ ( X , S ) is continuous.

For (1), by Theorem 4.1, we know for each A ∈ 2 ^X , it follows that S * ( A ) ⩾ S ( A ) . This implies that id X : ( X , S * ) ⟶ ( X , S ) is continuous.

For (2), for each M-fuzzifying convex space ( Y , C Y ) and each mapping φ : Y ⟶ X, define C X as in Lemma 4.2. Then ( X , C X ) is an M-fuzzifying convex space. Since φ : ( Y , C Y ) ⟶ ( X , S ) is continuous, we have ∀ A ∈ 2 X , C X ( A ) = C Y ( φ ← ( A ) ) ⩾ S ( A ) .

Further, by the definition of S * , we obtain S * ( A ) = ⋁ ⋃ dir j ∈ J A j = A ⋀ j ∈ J S ( A j ) ⩽ ⋁ ⋃ dir j ∈ J A j = A ⋀ j ∈ J C X ( A j ) ⩽ ⋁ ⋃ dir j ∈ J A j = A C X ( ⋃ dir j ∈ J A j ) = C X ( A ) = C Y ( φ ← ( A ) ) .

This proves that φ : ( Y , C Y ) ⟶ ( X , S * ) is CP, as desired. □

In a categorical sense, the category M-FYT of M-fuzzifying cotopological spaces is a topological category over Set. In the final, we will show that the M-FYC of M-fuzzifying convex spaces is also a topological category over Set.

Theorem 4.4. The category M-FYC is topological over Set.

Proof. We first prove the existence of final structures. Let ( ( X λ , C λ ) ) λ ∈ Λ be a family of M-fuzzifying convex spaces and let X be a nonempty set. Let further ( ( φ λ : ( X λ , C λ ) ⟶ X ) ) λ ∈ Λ be a sink. Define C : 2 X ⟶ M by C ( A ) = ⋀ λ ∈ Λ C λ ( φ λ ← ( A ) ) .

Since φ λ ← preserves arbitrary meets and directed joins, we can verify that C is an M-fuzzifying convex structure on X.

Let further ( Y , C Y ) be an M-fuzzifying convex space and ψ : X ⟶ Y be a mapping. Assume that ψ ∘ φ _λ is CP for all λ ∈ Λ. We have for all B ∈ 2 ^Y , C ( ψ ← ( B ) ) = ⋀ λ ∈ Λ C λ ( φ λ - 1 ( ψ ← ( B ) ) ) = ⋀ λ ∈ Λ C λ ( ( ψ ∘ φ λ ) ← ( B ) ) ) ⩾ ⋀ λ ∈ Λ C Y ( B ) = C Y ( B ) .

This implies that ψ : ( X , C ) ⟶ ( Y , C Y ) is CP, as desired.

Secondly, the class of all M-fuzzifying convex structures on a fixed set X is a subset of 2^{(M(2 X ))}, which means that the M-FYC fibre of X is aset.

Finally, for a one point set X = {x}, there exists only one M-fuzzifying convex structure C such that C ( { x } ) = ⊤ . Hence, M-FYC satisfies the terminal separator property. Therefore, M-FYC is a topological category in the sense of [27], that is, a well-fibred topological category in the terminology of [1]. □

5 Conclusions

In this paper, we studied the categorical relations between M-fuzzifying closure systems, M-fuzzifying cotopologies and M-fuzzifying convex structures. The category of M-fuzzifying cotopological spaces and the category of M-fuzzifying convex spaces are both bicoreflective subcategories of the category of M-fuzzifying closure spaces. Based on these conclusions, we know both M-fuzzifying cotopologies and M-fuzzifying convex structures can be treated as special cases of M-fuzzifying closure systems in a categorical sense.