A degree approach to separation axioms in M -fuzzifying convex spaces

Abstract

In this paper, S₀, S₁ and S₂ separation axioms are introduced in M-fuzzifying convex spaces. Each M-fuzzifying convex space can be regarded to be S₀, S₁ and S₂ separated to some degree. Some properties of them are investigated. The relations among them are discussed.

1 Introduction

The theory of convex sets, which is a branch of geometry, analysis and linear algebra, has numerous connections with other areas of mathematics. For different mathematical objects, there are so many collections of sets that can form convex structures, such as convexities in lattices and in Boolean algebras [25], convexities in real vector spaces [24], convexities in metric spaces [9], and convexities in graphs [22]. Also, convex structures appeared naturally in topology, especially in the theory of supercompact spaces [26]. As an independent spatial structure, convex structures are investigated from a topological aspect, such as product spaces, convex invariants and so on.

Separation axioms constitute one of the fundamental facets of the theory of convex structures. Jamison [3] introduced separation axioms and gave a restricted version of the polytope screening characterization in terms of screening with half-spaces. Kay and Womble [6] obtained characterizations of (finite) Carathéodory, Helly, and Radon numbers in terms of separation properties.

With the development of fuzzy mathematics, many mathematical structures have been generalized to the fuzzy case, such as fuzzy topology [1 , 32] and fuzzy convergence structures [11, 12]. Motivated by this, convex structures have also been endowed with fuzzy set theory. Weiss [27] considered a convex fuzzy set in a vector space over real or complex number. Katsaras and Liu [5] applied the concept of a fuzzy set to the elementary theory of vector spaces and topological vector spaces.

Maruyama [9] and Rosa [19] independently introduced the concept of fuzzy convex structures, which is called L-convex structures nowadays. Based on L-convex structures, Pang et al. presented several characterizations in a topological approach [10 , 18] and studied the spatial properties a categorical approach [13 , 17].

In a completely different direction, Shi and Xiu [20] provided a new approach to fuzzification of convex structures and proposed the notion of M-fuzzifying convex structures. Based on M-fuzzifying convex structures, Shi and Li [21] introduced the notion of M-fuzzifying restricted hull operators. Wu and Bai [28] proposed M-fuzzifying JHC convex structures and M-fuzzifying Peano interval spaces. Xiu et al. established its categorical relationship with M-fuzzifying closure systems [29] and proposed the concepts of M-fuzzifying interval operators [30] as well as axiomatic bases and axiomatic subbases [31].

In this paper, we will introduce the degree to which an M-fuzzifying convex space is S₀, S₁ and S₂ separated. A series of properties of them are investigated. Furthermore, we discuss the relations among them, and give a lot of examples to show the relations.

2 Preliminaries

Throughout this paper, (M, ∨, ∧, ′) is a completely distributive De Morgan algebra, i.e., a completely distributive lattice with an order-reversing involution ^′. 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, denoted by a ≺ b, if for every subset D ⊆ M, ⋁D ≥ b, there exists d ∈ D, d ≥ a [2]. A complete lattice M is completely distributive if and only if b =⋁ {a ∈ L| a ≺ b } for each b ∈ M. For a nonempty set X, 2^X denotes the powerset of X.

Definition 2.1. [25] A subset $C$ of 2^X is called a convex structure if it satisfies the following three conditions: (C1) $\emptyset, X \in C$ ; [(C2)] if ${A_{i} : i \in Ω} \subseteq C$ is nonempty, then $⋂_{i \in Ω} A_{i} \in C$ ; [(C3)] if ${A_{i} : i \in Ω} \subseteq C$ is nonempty and totally ordered by inclusion, then $⋃_{i \in Ω} A_{i} \in C$ .

The pair $(X, C)$ is called a convex space. The members of $C$ are called convex sets and their complements are called concave sets.

Definition 2.2. [25] Let $(X, C)$ be a convex space. A subset H of X is called a biconvex set (half-space) provided H is both convex set and convcave set.

Definition 2.3. [25] Let $(X, C)$ be a convex space and A ∈ 2^X. The (convex) hull of A is defined by $co (A) = \cap {C : A \subseteq C \in C} .$ A set of type co (F), with F finite, is called a polytope.

Definition 2.4. [4, 25] Let $(X, C)$ be a convex space.

(1) $(X, C)$ is said to be S₀ separated if for all x, y ∈ X with x ≠ y, then co ({x}) ≠ co ({y}); (2) $(X, C)$ is said to be S₁ separated if all singletons in X are convex; (3) $(X, C)$ is said to be S₂ separated if for all x, y ∈ X with x ≠ y, then there is a biconvex set H of X with x ∈ H, y ∉ H.

In [20], Shi and Xiu generalized the concept of convex structures to the concept of M-fuzzifying convex structures as follows.

Definition 2.5. [20] A mapping $C : 2^{X} \to M$ is called an M-fuzzifying convex structure on X if it satisfies the following conditions: (MYC1) $C (\emptyset) = C (X) = ⊤$ ; (MYC2) if {A_i : i ∈ Ω} ⊆2^X is nonempty, then $C (⋂_{i \in Ω} A_{i}) ⩾ ⋀_{i \in Ω} C (A_{i})$ ; (MYC3) if {A_i : i ∈ Ω} ⊆2^X is nonempty and totally ordered by inclusion, then $C (⋃_{i \in Ω} A_{i}) ⩾ ⋀_{i \in Ω} C (A_{i})$ .

If $C$ is an M-fuzzifying convex structure on X, then the pair $(X, C)$ is called an M-fuzzifying convex space. For A ∈ 2^X, $C (A)$ can be regarded as the degree to which A is a convex set.

Definition 2.6. [20] Let $(X, C)$ and $(Y, D)$ be M-fuzzifying convex spaces. A mapping f : X → Y is called an M-fuzzifying convexity preserving function if $C (f^{- 1} (B)) \geq D (B)$ for all B ∈ 2^Y.

Definition 2.7. [20] Let $(X, C)$ be an M-fuzzifying convex space and ∅ ≠ Y ⊆ X. $(Y, C |_{Y})$ is called the subspace of $(X, C)$ , where for each A ∈ 2^Y, $(C |_{Y}) (A) = ⋁ {C (B) : B \in 2^{X}, B \cap Y = A}$ .

Definition 2.8. [20] Let φ : 2^X → L be a mapping. The M-fuzzifying convex space $(X, C)$ generated by φ is given by

$\forall A \in 2^{X}, C (A) = ⋀ {D (A) : φ \leq D \in H},$

where $H$ denotes the family of all M-fuzzifying convex structures on X. Then φ is called a subbase of the M-fuzzifying convex structure $C$ . Alternatively, we say that φ generates the convex structure $C$ .

Definition 2.9. [20] Let ${(X_{t}, C_{t})}_{t \in T}$ be a family of M-fuzzifying convex spaces. Let X be the product of {X_t} _t∈T, and P_t : X → X_t denote the projection for each t ∈ T. Define a mapping φ : 2^X → L by $φ (A) = \underset{t \in T}{⋁} \underset{(P_{t})^{- 1} (B) = A}{⋁} C_{t} (B)$ for each A ∈ 2^X. Then the product M-fuzzifying convex structure $C$ on X is the one generated by the subbase φ. The resulting M-fuzzifying convex space $(X, C)$ is called the product of M-fuzzifying convex spaces ${(X_{t}, C_{t})}_{t \in T}$ and is denoted by $\prod_{t \in T} (X_{t}, C_{t})$ .

Theorem 2.10. [20] Let $(X, C)$ be the product of M-fuzzifying convex spaces ${(X_{t}, C_{t})}_{t \in T}$ . Then $C (\prod_{t \in T} A_{t})$ $\geq \underset{t \in T}{⋀} C_{t} (A_{t})$ .

Theorem 2.11. [20] Let $(X, C)$ be the product of M-fuzzifying convex spaces ${(X_{t}, C_{t})}_{t \in T}$ . Then for each t ∈ T, P_t : X → X_t is an M-fuzzifying convexity preserving function. Moreover, $C$ is the coarsest M-fuzzifying convex structure such that {P_t : t ∈ T} are M-fuzzifying convexity preserving functions.

3 S₀ separation axiom

In this section, we introduce the degree to which an M-fuzzifying convex space is S₀ separated.

Firstly, we investigate a charactration of S₀ separation axiom in convex space.

Lemma 3.1. A convex space $(X, C)$ is S₀ separated if and only if for any x, y ∈ X, there exists $A \in C$ such that x ∈ A, y ∉ A or $B \in C$ such that y ∈ B, x ∉ B.

Proof. For any x, y ∈ X with x ≠ y, we have

$\begin{matrix} co ({x}) \neq co ({y}) \\ \Leftrightarrow & y \notin co ({x}) or x \notin co ({y}) \\ \Leftrightarrow & there exists A \in C such that x \in A, y \notin A, or \\ B \in C such that y \in B, x \notin B . \end{matrix}$

Thus we complete the proof. □

Next, we generalize the degree to which an M-fuzzifying convex space is S₀ separated by using the above lemma.

Definition 3.2. For an M-fuzzifying convex space $(X, C)$ , we define the degree to which $(X, C)$ is S₀ separated as follows:

$S_{0} (X, C) = \underset{x \neq y}{⋀} (\underset{\binom{x \notin A}{y \in A}}{⋁} C (A) \lor \underset{\binom{x \in B}{y \notin B}}{⋁} C (B)) .$

Remark 3.3. If M is replaced by {0, 1}, then Definition 3.2 is equivalent to Definition 2.4 (1). So we can see that Definition 3.2 is reasonable generalization of S₀ separation axiom.

Theorem 3.4. Let $(X, C)$ be an M-fuzzifying convex space, and $(Y, C |_{Y})$ be the subspace of $(X, C)$ . Then $S_{0} (X, C) \leq S_{0} (Y, C |_{Y}) .$

Proof. Since $(C |_{Y}) (A) = ⋁ {C (B) : B \in 2^{X}, B \cap Y = A}$ , we know that $C (A) \leq C |_{Y} (A \cap Y), \forall A \in 2^{X} .$ Then $\begin{matrix} S_{0} (X, C) \\ = & \underset{\binom{x \neq y}{x, y \in X}}{⋀} (\underset{\binom{x \notin A}{y \in A}}{⋁} C (A) \lor \underset{\binom{x \in B}{y \notin B}}{⋁} C (B)) \\ \leq & \underset{\binom{x \neq y}{x, y \in Y}}{⋀} (\underset{\binom{x \notin A}{y \in A}}{⋁} C (A) \lor \underset{\binom{x \in B}{y \notin B}}{⋁} C (B)) \\ \leq & \underset{\binom{x \neq y}{x, y \in Y}}{⋀} (\underset{\binom{x \notin A}{y \in A}}{⋁} C |_{Y} (A \cap Y) \lor \underset{\binom{x \in B}{y \notin B}}{⋁} C |_{Y} (B \cap Y)) \\ = & \underset{\binom{x \neq y}{x, y \in Y}}{⋀} (\underset{\binom{x \notin A \cap Y}{y \in A \cap Y}}{⋁} C |_{Y} (A \cap Y) \lor \underset{\binom{x \in B \cap Y}{y \notin B \cap Y}}{⋁} C |_{Y} (B \cap Y)) \\ = & \underset{\binom{x \neq y}{x, y \in Y}}{⋀} (\underset{\binom{x \notin U}{y \in U}}{⋁} C |_{Y} (U) \lor \underset{\binom{x \in V}{y \notin V}}{⋁} C |_{Y} (V)) \\ = & S_{0} (Y, C |_{Y}) . \end{matrix}$

This means $S_{0} (X, C) \leq S_{0} (Y, C |_{Y}) .$ □

The above theorem can be interpreted as many-valued logical truth-value of the sentence “a subspace of an S₀ convex space is S₀". As we all know, the product of a family of S₀ convex space is S₀. Next we give the lattice-valued characterizations of this conclusion.

Theorem 3.5. Let $(X, C)$ be the product of M-fuzzifying convex spaces ${(X_{t}, C_{t})}_{t \in T}$ . Then

$\underset{t \in T}{⋀} S_{0} (X_{t}, C_{t}) \leq S_{0} (X, C) .$

Proof. Take any α ∈ M such that $\begin{matrix} α & ≺ & \underset{t \in T}{⋀} S_{0} (X_{t}, C_{t}) \\ = & \underset{t \in T}{⋀} \underset{\binom{x_{t} \neq y_{t}}{x_{t}, y_{t} \in X_{t}}}{⋀} (\underset{\binom{x_{t} \notin A_{t}}{y_{t} \in A_{t}}}{⋁} C_{t} (A_{t}) \lor \underset{\binom{x_{t} \in B_{t}}{y_{t} \notin B_{t}}}{⋁} C_{t} (B_{t})), \end{matrix}$ then for any t ∈ T, and x_t, y_t ∈ X_t with x_t ≠ y_t, we have $\begin{matrix} α & \leq & \underset{\binom{x_{t} \notin A_{t}}{y_{t} \in A_{t}}}{⋁} C_{t} (A_{t}) \lor \underset{\binom{x_{t} \in B_{t}}{y_{t} \notin B_{t}}}{⋁} C_{t} (B_{t}) \\ \leq & \underset{\binom{x_{t} \notin A_{t}}{y_{t} \in A_{t}}}{⋁} C (P_{t}^{- 1} (A_{t})) \lor \underset{\binom{x_{t} \in B_{t}}{y_{t} \notin B_{t}}}{⋁} C (P_{t}^{- 1} (B_{t})) . \end{matrix}$ For any $x = \prod_{t \in T} {x_{t}}, y = \prod_{t \in T} {y_{t}} \in X$ with x ≠ y, then there exists t₀ ∈ T such that x_{t
₀} ≠ y_{t
₀} . Then $\begin{matrix} α & \leq & \underset{\binom{x_{t_{0}} \notin A_{t_{0}}}{y_{t_{0}} \in A_{t_{0}}}}{⋁} C (P_{t_{0}}^{- 1} (A_{t_{0}})) \lor \underset{\binom{x_{t_{0}} \in B_{t_{0}}}{y_{t_{0}} \notin B_{t_{0}}}}{⋁} C (P_{t_{0}}^{- 1} (B_{t_{0}})) \\ = & \underset{\binom{x \notin P_{t_{0}}^{- 1} (A_{t_{0}})}{y \in P_{t_{0}}^{- 1} (A_{t_{0}})}}{⋁} C (P_{t_{0}}^{- 1} (A_{t_{0}})) \lor \underset{\binom{x \in P_{t_{0}}^{- 1} (B_{t_{0}})}{y \notin P_{t_{0}}^{- 1} (B_{t_{0}})}}{⋁} C (P_{t_{0}}^{- 1} (B_{t_{0}})) \\ \leq & \underset{\binom{x \notin A}{y \in A}}{⋁} C (A) \lor \underset{\binom{x \in B}{y \notin B}}{⋁} C (B) . \end{matrix}$ This implies that $α \leq \underset{\binom{x \neq y}{x, y \in X}}{⋀} (\underset{\binom{x \notin A}{y \in A}}{⋁} C (A) \lor \underset{\binom{x \in B}{y \notin B}}{⋁} C (B)) = S_{0} (X, C) .$ Since α is arbitrary, it follows that $\underset{t \in T}{⋀} S_{0} (X_{t}, C_{t}) \leq S_{0} (X, C)$ . □

4 S₁ separation axiom

In this section, we introduce the degree to which an M-fuzzifying convex space is S₁ separated, and discuss its relation with S₀ in M-fuzzifying convex space.

Definition 4.1. For an M-fuzzifying convex space $(X, C)$ , we define the degree to which $(X, C)$ is S₁ separated as follows: $S_{1} (X, C) = \underset{x \in X}{⋀} C ({x}) .$

Remark 4.2. If M is replaced by {0, 1}, then Definition 4.1 reduce to Definition 2.4 (2). So we can see that Definition 4.1 is reasonable generalization of S₁ separation axiom.

Theorem 4.3. Let $(X, C)$ be an M-fuzzifying convex space. Then

$S_{1} (X, C) = \underset{x \neq y}{⋀} \underset{\binom{x \in F}{y \notin F}}{⋁} C (F) .$

Proof. Firstly we prove that $S_{1} (X, C) \leq \underset{x \neq y}{⋀} \underset{\binom{x \in F}{y \notin F}}{⋁} C (F) .$ In fact, for any x, y ∈ X with x ≠ y, we have $C ({x}) \leq \underset{\binom{x \in F}{y \notin F}}{⋁} C (F) .$ This implies $S_{1} (X, C) = \underset{x \in X}{⋀} C ({x}) \leq \underset{x \neq y}{⋀} \underset{\binom{x \in F}{y \notin F}}{⋁} C (F) .$ Secondly we prove that $S_{1} (X, C) \geq \underset{x \neq y}{⋀} \underset{\binom{x \in F}{y \notin F}}{⋁} C (F) .$ Take any x ∈ X and α ∈ M such that $α ≺ \underset{x \neq y}{⋀} \underset{\binom{x \in F}{y \notin F}}{⋁} C (F)$ , we can obtain that for all y ∈ X with y ≠ x, there exists F_y with x ∈ F_y, y ∉ F_y, such that $α \leq C (F_{y}) .$ Since {x} = ⋂ _y≠xF_y, we obtain $α \leq \underset{y \neq x}{⋀} C (F_{y}) \leq C (⋂_{y \neq x} F_{y}) = C ({x}) .$ Hence $α \leq \underset{x \in X}{⋀} C ({x}) .$ Since α is arbitrary, we have $\underset{x \neq y}{⋀} \underset{\binom{x \in F}{y \notin F}}{⋁} C (F) \leq \underset{x \in X}{⋀} C ({x}) = S_{1} (X, C) .$ Therefore, $S_{1} (X, C) = \underset{x \neq y}{⋀} \underset{\binom{x \in F}{y \notin F}}{⋁} C (F) .$ □

By Theorem 4.3, we can obtain the following theorem.

Theorem 4.4. Let $(X, C)$ be an M-fuzzifying convex space. Then $S_{1} (X, C) \leq S_{0} (X, C) .$

Proof. It is straightforward and omitted. □

Example 4.5. Let X = {x, y} and $C : 2^{X} \to I$ be defined as follows: $C (A) = {\begin{matrix} 1 / 4, & A = {x}; \\ 1 / 5, & A = {y}; \\ 1, & A \in {\emptyset, X} . \end{matrix}$ Then $(X, C)$ is an I-fuzzifying convex space. From Definition 3.2 and 4.1, we have $S_{0} (X, C) = C ({x}) \lor C ({y}) = 1 / 4,$ and $S_{1} (X, C) = C ({x}) \land C ({y}) = 1 / 5 .$ Hence $S_{1} (X, C) < S_{0} (X, C) .$

Let $(X, C)$ be a crisp convex space. Then $C$ can be regarded as a mapping $χ_{C}$ : 2^X → M defined by $χ_{C} (A) = {\begin{matrix} ⊤, & A \in C; \\ ⊥, & A \notin C . \end{matrix}$

It is easy to prove that $(X, χ_{C})$ is a M-fuzzifying convex space.

Theorem 4.6. Let $(X, C)$ be a crisp convex space. Then $S_{1} (X, χ_{C}) = ⊤$ if and only if $(X, C)$ is S₁ separated.

Proof. It is straightforward and omitted. □

In the classical case, S₁ separation axiom has the heritability and the productive property. Next we will investigate these properties in M-fuzzifying convex spaces.

Theorem 4.7. Let $(X, C)$ be an M-fuzzifying convex space, and $(Y, C |_{Y})$ be the subspace of $(X, C)$ . Then

$S_{1} (X, C) \leq S_{1} (Y, C |_{Y}) .$

Proof. Since $C |_{Y} (A) = ⋁ {C (B) : B \in 2^{X}, B \cap Y = A},$ we have $\begin{matrix} S_{1} (X, C) & = & \underset{x \in X}{⋀} C ({x}) \\ \leq & \underset{y \in Y}{⋀} C ({y}) \\ \leq & \underset{y \in Y}{⋀} ⋁ {C (B) : B \in 2^{X}, B \cap Y = {y}} \\ = & \underset{y \in Y}{⋀} C |_{Y} ({y}) = S_{1} (Y, C |_{Y}) . \end{matrix}$ This implies $S_{1} (X, C) \leq S_{1} (Y, C |_{Y}) .$ □

Theorem 4.8. Let $(X, C)$ be the product of M-fuzzifying convex spaces ${(X_{t}, C_{t})}_{t \in T}$ . Then $\underset{t \in T}{⋀} S_{1} (X_{t}, C_{t}) \leq S_{1} (X, C) .$

Proof. Take any α ∈ M such that $α ≺ \underset{t \in T}{⋀} S_{1} (X_{t}, C_{t}),$ then for any t ∈ T, we know $α \leq S_{1} (X_{t}, C_{t}) = \underset{x_{t} \in X_{t}}{⋀} C_{t} ({x_{t}}) .$ Therefore, $α \leq C_{t} ({x_{t}}), \forall t \in T, x_{t} \in X_{t} .$ Take any $x = \prod_{t \in T} {x_{t}} \in X,$ by Theorem 2.10, we can obtain $C ({x}) = C (\prod_{t \in T} {x_{t}}) \geq \underset{t \in T}{⋀} C_{t} ({x_{t}}) \geq α .$ Hence $α \leq \underset{x \in X}{⋀} C ({x}) = S_{1} (X, C) .$ Since α is arbitrary, it follows that $\underset{t \in T}{⋀} S_{1} (X_{t}, C_{t}) \leq S_{1} (X, C)$ . □

5 S₂ separation axiom

In this section, we introduce the degree to which an M-fuzzifying convex space is S₂ separated, and discuss its relation with S₁ in M-fuzzifying convex space.

Firstly, we introduce the degree to which a set is biconvex set in M-fuzzifying convex spaces.

Definition 5.1. Let $(X, C)$ be an M-fuzzifying convex space and A ∈ 2^X. The degree $H_{C} (A)$ to which A is biconvex set (half-space, hemispace) is defined by $H_{C} (A) = C (A) \land C (A^{c}) .$

Remark 5.2. If M is replaced by {0, 1}, then Definition 5.1 reduce to Definition 2.2. So we can see that Definition 5.1 is reasonable generalization of biconvex set.

Theorem 5.3. Let $(X, C)$ and $(Y, D)$ be M-fuzzifying convex spaces, $f : (X, C) \to (Y, D)$ be M-fuzzifying convex preserving function. Then $H_{D} (A) \leq H_{C} (f^{- 1} (A)) .$

Proof. By Definition 2.6, we have $\begin{matrix} H_{D} (A) & = & D (A) \land D (A^{c}) \\ \leq & C (f^{- 1} (A)) \land C (f^{- 1} (A^{c})) \\ = & C (f^{- 1} (A)) \land C ((f^{- 1} (A))^{c}) \\ = & H_{C} (f^{- 1} (A)) . \end{matrix}$ Thus we complete the proof. □

Theorem 5.4. Let $(X, C)$ be an M-fuzzifying convex space, $(Y, C |_{Y})$ be the subspace of $(X, C)$ , A ⊆ X. Then $H_{C} (A) \leq H_{D} (A \cap Y) .$

Proof. From $\begin{matrix} C |_{Y} (A \cap Y) & = & ⋁ {C (B) : B \in 2^{X}, B \cap Y = A \cap Y} \\ ⩾ & C (A), \end{matrix}$ and $\begin{matrix} C |_{Y} (Y \ (A \cap Y)) \\ = & C ((X \ A) \cap Y) \\ = & ⋁ {C (B) : B \in 2^{X}, B \cap Y = (X \ A) \cap Y} \\ ⩾ & C (X \ A), \end{matrix}$ we can obtain $C (A) \land C (X \ A) \leq C |_{Y} (A \cap Y) \land C |_{Y} (Y \ (A \cap Y)) .$ Therefore, $H_{C} (A) \leq H_{C |_{Y}} (A \cap Y) .$ Thus we complete the proof. □

Next we introduce the degree to which an M-fuzzifying convex space is S₂ separated by using biconvex set.

Definition 5.5. For an M-fuzzifying convex space $(X, C)$ , we define the degree to which $(X, C)$ is S₂ separated as follows: $S_{2} (X, C) = \underset{x \neq y}{⋀} ⋁ {H_{C} (A) : A \in 2^{X}, x \in A, y /\in A} .$

Remark 5.6. If M is replaced by {0, 1}, then Definition 5.5 reduce to Definition 2.4 (3). So we can see that Definition 5.5 is reasonable generalization of S₂ separation axiom.

Theorem 5.7. Let $(X, C)$ be a crisp convex space. Then $S_{2} (X, χ_{C}) = ⊤$ if and only if $(X, C)$ is S₂ separated.

Proof. $\begin{matrix} S_{2} (X, χ_{C}) = ⊤ \\ \Leftrightarrow & for all x and y with x \neq y, there exists A \in 2^{X} \\ with x \in A, y \notin A, such that H_{χ_{C}} (A) = χ_{C} (A) \\ \land χ_{C} (A^{c}) = ⊤ \\ \Leftrightarrow & for all x and y with x \neq y, there exists A \in 2^{X} \\ with x \in A, y \notin A, such that χ_{C} (A) = ⊤, and \\ χ_{C} (A^{c}) = ⊤ \\ \Leftrightarrow & for all x and y with x \neq y, there exists A \in 2^{X} \\ with x \in A, y \notin A, such that A, A^{c} \in C \\ \Leftrightarrow & for all x and y with x \neq y, there exists A \in 2^{X} \\ with x \in A, y \notin A, such that A is a biconvex set \\ in (X, C) \\ \Leftrightarrow & (X, C) is S_{2} separated . \end{matrix}$ Thus we complete the proof. □

Theorem 5.8. Let $(X, C)$ be an M-fuzzifying convex space. Then $S_{2} (X, C) \leq S_{1} (X, C) .$

Proof. Let α ∈ M such that $\begin{matrix} α & ≺ & S_{2} (X, C) \\ = & \underset{x \neq y}{⋀} ⋁ {H_{C} (A) : A \in 2^{X}, x \in A, y \notin A}, \end{matrix}$ and x ∈ X . For any y ∈ X with y ≠ x, there exists A_y ∈ 2^X with y ∈ A_y and x ∉ A_y such that $α \leq H_{C} (A_{y}) = C (A_{y}) \land C ({A_{y}}^{c}) .$ Since X ∖ {x} = ⋃ _y≠xA_y, we have $\begin{matrix} α & \leq & \underset{y \neq x}{⋀} (C (A_{y}) \land C ({A_{y}}^{c})) \\ = & (\underset{y \neq x}{⋀} C (A_{y})) \land (\underset{y \neq x}{⋀} C ({A_{y}}^{c})) \\ \leq & C (⋂_{y \neq x} A_{y}) \land C (⋂_{y \neq x} ({A_{y}}^{c})) \\ = & C (⋂_{y \neq x} A_{y}) \land C ({(⋃_{y \neq x} A_{y})}^{c}) \\ = & C (⋂_{y \neq x} A_{y}) \land C ({x}) \\ \leq & C ({x}) . \end{matrix}$ Therefore, $α \leq \underset{x \in X}{⋀} C ({x}) = S_{1} (X, C) .$ Since α is arbitrary, we have $S_{2} (X, C) \leq S_{1} (X, C) .$ □

Example 5.9. Let X = {x, y, z} and $C : 2^{X} \to I$ be defined as follows: $C (A) = {\begin{matrix} 6 / 10, & A \in {{X, z}, {Y, z}}; \\ 7 / 10, & A \in {{Y}, {z}}; \\ 8 / 10, & A \in {{X}}; \\ 9 / 10, & A \in {{X, Y}}; \\ 1, & A \in {\emptyset, X} . \end{matrix}$ Then $(X, C)$ is an I-fuzzifying convex space. From Definition 4.1 and 5.5, we have $S_{1} (X, C) = C ({x}) \land C ({y}) \land C ({z}) = 7 / 10,$ and $\begin{matrix} S_{2} (X, C) & = & \underset{\binom{x \neq y}{x, y \in X}}{⋀} ⋁ {H_{C} (A) : A \in 2^{X}, x \in A, y \notin A} \\ = & 6 / 10 . \end{matrix}$ This imples $S_{2} (X, C) < S_{1} (X, C) .$

Example 5.10. In Example 4.5, $S_{1} (X, C) = 1 / 5 .$ From Definition 5.5, we can obtain $S_{2} (X, C) = C ({x}) \land C ({y}) = 1 / 5 .$ Hence, $S_{2} (X, C) = S_{1} (X, C) .$

In the classical case, S₂ separation axiom has the heritability and the productive property. Next we will investigate these properties in M-fuzzifying convex spaces.

Theorem 5.11. Let $(X, C)$ be an M-fuzzifying convex space, and $(Y, C |_{Y})$ be the subspace of $(X, C)$ . Then $S_{2} (X, C) \leq S_{2} (Y, C |_{Y}) .$

Proof. Since $C (A) \leq C |_{Y} (A \cap Y), \forall A \in 2^{X}$ , we know that $\begin{matrix} S_{2} (X, C) \\ = & \underset{\binom{x, y \in X}{x \neq y}}{⋀} ⋁ {H_{C} (A) : A \in 2^{X}, x \in A, y \notin A} \\ \leq & \underset{\binom{x, y \in Y}{x \neq y}}{⋀} ⋁ {H_{C} (A) : A \in 2^{X}, x \in A, y \notin A} \\ = & \underset{\binom{x, y \in Y}{x \neq y}}{⋀} ⋁ {C (A) \land C (X ∖ A) : A \in 2^{X}, x \in A, y \notin A} \\ \leq & \underset{\binom{x, y \in Y}{x \neq y}}{⋀} ⋁ {C |_{Y} (A \cap Y) \land C |_{Y} ((X ∖ A) \cap Y) : \\ A \in 2^{X}, x \in A, y \notin A} \\ = & \underset{\binom{x, y \in Y}{x \neq y}}{⋀} ⋁ {C |_{Y} (A \cap Y) \land C |_{Y} (Y ∖ (A \cap Y)) : \\ A \in 2^{X}, x \in A, y \notin A} \\ = & \underset{\binom{x, y \in Y}{x \neq y}}{⋀} ⋁ {H_{C |_{Y}} (A \cap Y) : A \in 2^{X}, x \in A, y \notin A} \\ = & \underset{\binom{x, y \in Y}{x \neq y}}{⋀} ⋁ {H_{C |_{Y}} (A \cap Y) : A \in 2^{X}, x \in A \cap Y, \\ y \notin A \cap Y} \\ = & \underset{\binom{x, y \in Y}{x \neq y}}{⋀} ⋁ {H_{C |_{Y}} (B) : B \in 2^{Y}, x \in B, y \notin B} \\ = & S_{2} (Y, C |_{Y}) . \end{matrix}$ Thus we complete the proof. □

Theorem 5.12. Let $(X, C)$ be the product of M-fuzzifying convex spaces ${(X_{t}, C_{t})}_{t \in T}$ . Then $\underset{t \in T}{⋀} S_{2} (X_{t}, C_{t}) \leq S_{2} (X, C) .$

Proof. Take any α ∈ M such that $\begin{matrix} α & ≺ & \underset{t \in T}{⋀} S_{2} (X_{t}, C_{t}) \\ = & \underset{t \in T}{⋀} \underset{x_{t} \neq y_{t}}{⋀} ⋁ {H_{C_{t}} (A_{t}) : A_{t} \in 2^{X_{t}}, x_{t} \in A_{t}, y_{t} \notin A_{t}}, \end{matrix}$ then for any t ∈ T, and x_t, y_t ∈ X_t with x_t ≠ y_t, there exists A_t ∈ 2^{X
_t} with x_t ∈ A_t, y_t ∉ A_t, such that $α \leq H_{C_{t}} (A_{t}) .$ (1) For any $x = \prod_{t \in T} {x_{t}}, y = \prod_{t \in T} {y_{t}} \in X$ with x ≠ y , then there exists t₀ ∈ T such that x_{t
₀} ≠ y_{t
₀} . By (1), we know that there exists A_{t
₀} ⊆ X_{t
₀} with x_{t
₀} ∈ A_{t
₀}, y_{t
₀} ∉ A_{t
₀} such that $\begin{matrix} α & \leq & H_{C_{t_{0}}} (A_{t_{0}}) \\ = & C_{t_{0}} (A_{t_{0}}) \land C_{t_{0}} (X_{t_{0}} ∖ A_{t_{0}}) \\ \leq & C ({P_{t_{0}}}^{- 1} (A_{t_{0}})) \land C ({P_{t_{0}}}^{- 1} (X_{t_{0}} ∖ A_{t_{0}})) \\ = & C ({P_{t_{0}}}^{- 1} (A_{t_{0}})) \land C (X ∖ {P_{t_{0}}}^{- 1} (A_{t_{0}})) \\ = & H_{C} ({P_{t_{0}}}^{- 1} (A_{t_{0}})) . \end{matrix}$ Since x_{t
₀} ∈ A_{t
₀}, y_{t
₀} ∉ A_{t
₀}, we know that x ∈ P_{t
₀}^-1 (A_{t
₀}) , y ∉ P_{t
₀}^-1 (A_{t
₀}) . Hence $α \leq ⋁ {H_{C} (A) : A \in 2^{X}, x \in A, y \notin A} .$ Therefore $\begin{matrix} α & \leq & \underset{x \neq y}{⋀} ⋁ {H_{C} (A) : A \in 2^{X}, x \in A, y \notin A} \\ = & S_{2} (X, C) . \end{matrix}$ Since α is arbitrary, it follows that $⋀_{t \in T} S_{2} (X_{t}, C_{t}) \leq S_{2} (X, C)$ . □

6 Conclusions

The theory of fuzzy convex structures is a new branch of fuzzy mathematics. In this paper, we firstly introduced the degree to which an M-fuzzifying convex space is S₀, S₁ and S₂ separated. Each M-fuzzifying convex space can be regarded to be S₀, S₁ and S₂ separated to some degree. We discussed the relations among them, and give a lot of examples to show the relations. Also, we investigated the heritability and the productive property of S₀, S₁ and S₂ separation axioms in M-fuzzifying convex spaces.

Footnotes

Acknowledgement

The authors would like to express their sincere thanks to the anonymous reviewers for their careful reading and constructive comments. This work is supported by Beijing Natural Science Foundation (1174014), the Outstanding Talents Program of Beijing (2016000020124G017).

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A degree approach to separation axioms in M -fuzzifying convex spaces

Abstract

1 Introduction

2 Preliminaries

3 S0 separation axiom

4 S1 separation axiom

5 S2 separation axiom

Footnotes

Acknowledgement

References

3 S₀ separation axiom

4 S₁ separation axiom

5 S₂ separation axiom