r -Near topologies on nearness approximation spaces

1 Introduction

In this section, we give properties and some definitions of near sets defined by Peters [5, 6].

Object Description

Objects are known with their definitions. For instance, when we say the definition of x ∈ O object, we mean Φ (x) functional value sequence. The important thing here is the selection of necessary probe function φ i : O → ℝ to define object. Here, the less the given functions quantity the weaker the definition of the object. We can give the closest definition of the object by multiplying the functions. Let B ⊆ F be the set of functions which presents the description of the object. Then φ _i functions over set B give us a close definition about the object. So, if we define a vector valued function such as Φ (x) = (φ₁ (x) , φ₂ (x) , . . . , φ _L  (x)), this will be a basis for the definition of the object.

Nearness of Object

Definition 1. Let x, x′ ∈ O and B ⊆ F. Then

_{∼ B} = {(x, x′) ∈ O × O : Δ _{φ i}  = 0, ∀φ _i  ∈ B} is called the indiscernibility relation on O, where the description length i ≤ |Φ|.

In the equivalence relation given in the above, it can be said that all the definitions of the objects x and x′ in the same equivalence class are the same, that is, the objects are very near to each other or even the same.

Definition 2. [5] Let B ⊆ F be a set of functions representing the features of objects x, x′ ∈ O. Objects x, x′ are called minimally near each other if there exists φ _i  ∈ B such that x_{∼  {φ i }} x′, Δ _{φ i}  = 0. We call it the Nearness Description Principle (NDP).

Fundamental Approximation space

Definition 3. [5] Let O be a set of perceptual objects, F be a set of functions representing object features, B ⊆ F and _{∼ B}be indiscernibility relation. Then the triple FAS = (O, F, _{∼ B}) is called Fundamental Approximation Space.

Definition 4. [5] Let (O, F, _{∼ B}) be a Fundamental Approximation Space and A ⊆ O.

(1) The set of union of [x]  _B  ∈ O/_{∼ B} which is subset of A, is called B lower approximation of A and denoted as

B ∗ A = ∪ [ x ] B ⊆ A [ x ] B .

(2) The set of union of [x]  _B  ∈ O/_{∼ B} elements, whose intersection with A is non-empty, is called B upper approximation of A and defined as

B ∗ A = ∪ [ x ] B ∩ A ≠ ∅ [ x ] B .

(3) The boundary of A is denoted as Bnd _B A and defined as

Bnd _B A = B^∗A ∖ B_∗A = {x ∈ O : x ∈ B^∗A and x ∉ B_∗A}.

Nearness Approximation Space

Indiscernibility relation can be defined for each subset B _r , such that B _r  ⊆ B ⊆ F and |B _r | = r. Let us denote that relation with _{∼ B r} . _{∼ B r} can form different decomposition for each r over O. Here, _{∼ B r} separate O to [x]  _{B r} nearness classes and O/_{∼ B r}  = {[x]  _{B r}  : x ∈ O} set is quotient set. Consequently, N _r  (B) = {O/_{∼ B r}  : B _r  ⊆ B} set of all O/_{∼ B r} partitions is obtained.

Definition 5. [5] Let O be a set of perceptual objects, F be a set of functions representing object features and B ⊆ F. Then quadruple of NAS = (O, F, _{∼ B r} , N _r  (B)) is called Nearness Approximation Space.

Definition 6. [5] Let (O, F, _{∼ Br} , N _r  (B)) be a Nearness Approximation Space and A ⊆ O.

(1) The union of [x]  _{B r}  ∈ O/_{∼ B r} elements, whose are subset of A, is called B _r lower Approximation of A and defined as

N r ( B ) ∗ A = ∪ [ x ] B r ⊆ A [ x ] B r .

(2) The union of [x]  _{B r}  ∈ O/_{∼ B r} elements, whose intersection with A is non-empty, is called B _r upper Approximation of A and defined as

N r ( B ) ∗ A = ∪ [ x ] B r ∩ A ≠ ∅ [ x ] B r .

The Rough Set theory proposed by Pawlak consists of the upper and lower approximation of a set and is obtained with the help of a relation that matches the properties of the object called indiscernibility relation. The near set theory is a more general version of the mentioned set theory. This theory is also based on the indiscernibility relation, but here the indiscernibility relation is considered not for all the functions that give the object properties but separately for each subset. In this way, the partitions of the set of objects are obtained. According to the number of functions (r), N _r  (B) sets are formed by the combination of these partitions. Similar to the rough set theory, lower and upper approximations are obtained based on the N _r  (B) set. Therefore, unlike the equality in the rough set theory, this system, created in the near sets, is to characterize the number of common properties of objects. The increase in the number of these common properties increases the nearness of objects to each other. Also, if we use all the functions that give the object properties, the approaches in the near set and those in the rough set coincide.

2 r-Near topological spaces

Definition 7. Let (O, F, _{∼ Br} , N _r  (B)) be a Nearness Approximation Space, X ⊆ O and (X, τ) be a topological space. The family τ r ∗ = { N r ( B ) ∗ ( G ) : G ∈ τ } is called r-near topology which generated by (O, F, _{∼ Br} , N _r  (B)). The elements N _r  (B) ^∗ (G) are called r-near open sets. Complement of r-near open sets are called r-near closed sets.

Throughout the paper, N _r  (B) ^∗ (G) r-near open sets are denoted as G r ∗ for simplicity unless specified otherwise. The family of all r-near closed sets is denoted as τ r ∗ k .

Example 1. Let X = {a, b, c, d, e, f}, B = {φ₁, φ₂, φ₃} a set of functions representing object features and τ = {∅ , X, {c} , {a, b, c} , {c, d, e, f}}. Then function φ _i defined as

Let us construct the equivalence classes for each r-combination. These equivalence classes are formed as

[a] _{φ1} = {a, c, e}, [b] _{φ1} = {b, d, f}

[a] _{φ2} = {a, d}, [b] _{φ2} = {b, e, f}, [c] _{φ2} = {c}

[a] _{φ3} = {a, e}, [b] _{φ3} = {b, f}, [c] _{φ3} = {c}, [d] _{φ3} = {d},

[a] _{φ1,φ2} = {a}, [b] _{φ1,φ2} = {b, f}, [c] _{φ1,φ2} = {c} , [d] _{φ1,φ2} = {d}, [e] _{φ1,φ2} = {e}

[a] _{φ1,φ3} = {a, e}, [b] _{φ1,φ3} = {b, f}, [c] _{φ1,φ3} = {c}, [d] _{φ1,φ3} = {d}

[a] _{φ2,φ3} = {a}, [b] _{φ2,φ3} = {b, f}, [c] _{φ2,φ3} = {c}, [d] _{φ2,φ3} = {d}, [e] _{φ2,φ3} = {e}

[a] _{{φ1,φ2,φ3}} = {a}, [b] _{{φ1,φ2,φ3}} = {b, f}, [c] _{{φ1,φ2,φ3}} = {c}, [d] _{{φ1,φ2,φ3}} = {d}, [e] _{{φ1,φ2,φ3}} = {e}.

Where the set of equivalence classes for each r-combination are formed as

N₁ (B) = {[a] _{φ1}, [b] _{φ1}, [a] _{φ2}, [b] _{φ2}, [c] _{φ2}, [a] _{φ3}, [b] _{φ3}, [c] _{φ3}, [d] _{φ3}}

N₂ (B) = {[a] _{φ1,φ2}, [b] _{φ1,φ2}, [c] _{φ1,φ2}, [d] _{φ1,φ2}, [e] _{φ1,φ2}, [a] _{φ1,φ3}, [b] _{φ1,φ3}, [c] _{φ1,φ3}, [d] _{φ1,φ3}, [a] _{φ2,φ3}, [b] _{φ2,φ3}, [c] _{φ2,φ3}, [d] _{φ2,φ3}, [e] _{φ2,φ3}}

N₃ (B) = {[a] _{{φ1,φ2,φ3}}, [b] _{{φ1,φ2,φ3}}, [c] _{{φ1,φ2,φ3}}, [d] _{{φ1,φ2,φ3}}, [e] _{{φ1,φ2,φ3}}}. Let us now construct the τ r ∗ r-near topology. Where

and τ 1 ∗ = { ∅ , X , { a , c , e } } . Similiarly τ 2 ∗ = { ∅ , X , { c } , { a , b , c , e , f } } and τ 3 ∗ = { ∅ , X , { c } , { a , b , c , f } , { b , c , d , e , f } } .

As seen in the Example 2, even-though τ 1 ∗ and τ 2 ∗ families are topology, τ 3 ∗ family is not a topology. Let us show which of the terms of being topology that τ r ∗ families satisfy.

Remark 1. Notice that the N₁ (B) and N₂ (B) families are not partitions of O. However, the N₃ (B) family is a partition. This confusion arises from the selection of all probe functions. Moreover, this is the main relationship between the Rough sets and Near sets that Peters mentioned [5].

Theorem 1. Let (O, F, _{∼ Br} , N _r  (B)) be a Nearness Approximation Space, X ⊆ O and (X, τ) be a topological space. Then ∅ and X are r-near open sets.

Proof. This is obvious from ∅ r ∗ = ∅ and X r ∗ = X .

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Theorem 2. Let (O, F, _{∼ Br} , N _r  (B)) be a Nearness Approximation Space, X ⊆ O and (X, τ) be a topological space. If ( G i ) r ∗ ∈ τ r ∗ , for all i ∈ I, then ∪ i ∈ I ( G i ) r ∗ ∈ τ r ∗ , where I is an arbitrary index set.

Proof. Let ( G i ) r ∗ ∈ τ r ∗ , for all i ∈ I. Then

( G i ) r ∗ = ∪ [ x ] B r ∩ G i ≠ ∅ [ x ] B r . On the other hand,

∪ i ∈ I ( G i ) r ∗ = ∪ i ∈ I ( ∪ [ x ] B r ∩ G i ≠ ∅ [ x ] B r ) = ∪ [ x ] B r ∩ ( ∪ i ∈ I G i ) ≠ ∅ [ x ] B r . Since τ is a topology, ∪ i ∈ I G i ∈ τ , that is ∪ i ∈ I ( G i ) r ∗ ∈ τ r ∗ .■

Definition 8. [11] A subclass τ^∗ ⊂ P (X) is called supra topology on X if X ∈ τ^∗ and τ^∗ closed under arbitrary union.

Corollary 1. Let (O, F, _{∼ Br} , N _r  (B)) be a Nearness Approximation Space, X ⊆ O and (X, τ) be a topological space. The families τ r ∗ are supra topology on X.

Theorem 3. Let (O, F, _{∼ Br} , N _r  (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and r ≤ s ≤ |B|. Then followings are true;

(1) If G ⊆ H, then G r ∗ ⊆ H r ∗ .

(2) If B _r  ⊆ B_s , then [x]  _{B s}  ⊆ [x]  _{B r} .

(3) G ⊆ G r ∗ , for all G ∈ τ.

(4) G s ∗ ⊆ G r ∗ , for all G ∈ τ.

(5) If G ∈ τ and x₀ ∈ G, then G r ∗ is a neighborhood of x₀.

Proof. (1) Let G ⊆ H. Then G r ∗ = ∪ [ x ] B r ∩ G ≠ ∅ [ x ] B r ⊆ ∪ [ x ] B r ∩ H ≠ ∅ [ x ] B r = H r ∗ .

(2)

[ x ] B r = { x ′ ∈ X : x ∽ B r x ′ } = { x ′ ∈ X : { ( x , x ′ ) : φ ( x ) = φ ( x ′ ) , ∀ φ ∈ B r } ⊆ { x ′ ∈ X : { ( x , x ′ ) : φ ( x ) = φ ( x ′ ) , ∀ φ ∈ B s } = { x ′ ∈ X : x ∽ B s x ′ } = [ x ] B s

(3) G = ∪ x ∈ G { x } ⊆ ∪ [ x ] B r ∩ G ≠ ∅ [ x ] B r = G r ∗

(4) G r ∗ = ∪ [ x ] B r ∩ G ≠ ∅ [ x ] B r ⊆ ∪ [ x ] B s ∩ G ≠ ∅ [ x ] B s = G s ∗

(5) Obvious from (3).■

Definition 9. Let (O, F, _{∼ Br} , N _r  (B)) be a Nearness Approximation Space, X ⊆ O and (X, τ) be a topological space. If there exists a G r ∗ ∈ τ r ∗  such that x ∈ G ⊆ G r ∗ ⊆ N , then N is called r-near neighborhood of x. If N r-near open set, then N is called r-near open neighborhood.

The family of all r-near neighborhood of x is denoted N r ∗ ( x ) and the family of all r-near open neighborhood is denoted as τ r ∗ ( x ) .

Example 2. If we consider Nearness Approxi-mation Space in Example 2 and τ 2 ∗ = { ∅ , X , { c } , { a , b , c , e , f } } . Then N 2 ∗ ( b ) = { { a , b , c , e , f } , X } and τ 2 ∗ ( c ) = { X , { c } , { a , b , c , e , f } } .

Proposition 1. Let (O, F, _{∼ Br} , N _r  (B)) be a Nearness Approximation Space, X ⊆ O and (X, τ) be a topological space. Then followings are true;

(1) If N ∈ N r ∗ ( x ) , then x ∈ N.

(2) If N 1 ∈ N r ∗ ( x ) and N₁ ⊆ N₂, then N 2 ∈ N r ∗ ( x ) .

(3) If N₁, N 2 ∈ N r ∗ ( x ) , then N 1 ∩ N 2 ∈ N r ∗ ( x ) .

Proof. (1) Let N ∈ N r ∗ ( x ) . Then there exists a G r ∗ ∈ τ r ∗ such that x ∈ G ⊆ G r ∗ ⊆ N . Therefore, x ∈ N.

(2) Let N 1 ∈ N r ∗ ( x ) and N₁ ⊆ N₂. Then there exists a G r ∗ ∈ τ r ∗ such that x ∈ G ⊆ G r ∗ ⊆ N 1 ⊆ N 2 . Therefore, N 2 ∈ N r ∗ ( x ) .

(3) Let N₁, N 2 ∈ N r ∗ ( x ) . Then there exist G r ∗ , H r ∗ ∈ τ r ∗ such that x ∈ G ⊆ G r ∗ ⊆ N 1 and x ∈ H ⊆ H r ∗ ⊆ N 2 . Since τ is a topology, G ∩ H ∈ τ, x ∈ G ∩ H ⊆ ( G ∩ H ) r ∗ ⊆ G r ∗ ⊆ N 1 and x ∈ G ∩ H ⊆ ( G ∩ H ) r ∗ ⊆ H r ∗ ⊆ N 2 . Consequently, x ∈ G ∩ H ⊆ ( G ∩ H ) r ∗ ⊆ N 1 ∩ N 2 .■

Proposition 2. Let (O, F, _{∼ Br} , N _r  (B)) be a Nearness Approximation Space, X ⊆ O and (X, τ) be a topological space. If U neighborhood of  x, then N _r  (B) ^∗ (U) is also a r-near neighborhood of x.

Proof. Let U be neighborhood of x. Then there exists a G ∈ τ such that x ∈ G ⊆ U. On the other hand, since G r ∗ = N r ( B ) ∗ ( G ) = ∪ [ x ] B r ∩ G ≠ ∅ [ x ] B r ⊆ G ⊆ U ∪ [ x ] B r ∩ U ≠ ∅ [ x ] B r = N r ( B ) ∗ ( U ) , x ∈ G ⊆ G r ∗ ⊆ N r ( B ) ∗ ( U ) . This completes the proof.■

Theorem 4. Let (O, F, _{∼ Br} , N _r  (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and r ≤ s ≤ |B|. If N ∈ N r ∗ ( x 0 ) for all x₀ ∈ X, then N ∈ N s ∗ ( x 0 ) .

Proof. Let N ∈ N r ∗ ( x 0 ) . Then there exists a G r ∗ ∈ τ r ∗ such that x 0 ∈ G ⊆ G r ∗ ⊆ N . Since G s ∗ ⊆ G r ∗ by Theorem 2 (4), x 0 ∈ G ⊆ G s ∗ ⊆ G r ∗ ⊆ N . This completes the proof.■

Definition 10. Let (O, F, _{∼ Br} , N _r  (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and A ⊆ X.

(1) The set cl r ∗ A = ∩ { F r ∗ : F r ∗ is a r-near closed set and A ⊆ F r ∗ } is called r-near closure of A .

(2) The set int r ∗ A = ∪ { G r ∗ : G r ∗ is a r-near open set and G r ∗ ⊆ A } is called r-near interior of A .

Example 3. Let us consider the Nearness Approximation Space and τ 3 ∗ = { ∅ , X , { c } , {a, b, c, f} , {b, c, d, e, f}} in Example 2. If A = {a, b, c, d, f}, then the interior of A is int r ∗ A = { a , b , c , f } and the closure of A is cl r ∗ A = X .

Theorem 5. Let (O, F, _{∼ Br} , N _r  (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and A, B ⊆ X. Then the followings are true;

(1) If A ⊆ B, then cl r ∗ A ⊆ cl r ∗ B .

(2) If A ⊆ B, then int r ∗ A ⊆ int r ∗ B .

Proof. (1)

A ⊆ B ⇒ { F r ∗ : F r ∗ ∈ τ r ∗ k , B ⊆ F r ∗ } ⊆ { F r ∗ : F r ∗ ∈ τ r ∗ k , A ⊆ F r ∗ } ⇒ ∩ { F r ∗ : F r ∗ ∈ τ r ∗ k , A ⊆ F r ∗ } ⊆ ∩ { F r ∗ : F r ∗ ∈ τ r ∗ k , B ⊆ F r ∗ } ⇒ cl r ∗ A ⊆ cl r ∗ B

(2) A ⊆ B ⇒ { G r ∗ : G r ∗ ∈ τ r ∗ , G r ∗ ⊆ A } ⊆ { G r ∗ : G r ∗ ∈ τ r ∗ , G r ∗ ⊆ B } ⇒ ∪ { G r ∗ : G r ∗ ∈ τ r ∗ , G r ∗ ⊆ A } ⊆ ∪ { G r ∗ : G r ∗ ∈ τ r ∗ , G r ∗ ⊆ B } ⇒ int r ∗ A ⊆ int r ∗ B

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Proposition 3. Let (O, F, _{∼ Br} , N _r  (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and A ⊆ X. Then the followings are true;

(1) X ∖ int r ∗ A = cl r ∗ ( X ∖ A )

(2) X ∖ cl r ∗ A = int r ∗ ( X ∖ A )

Proof. (1)

X ∖ int r ∗ A = X ∖ ∪ { G r ∗ : G r ∗ is r - near open set and G r ∗ ⊆ A } = ∩ { X ∖ G r ∗ : G r ∗ r set and G r ∗ ⊆ A } = ∩ { F r ∗ : F r ∗ r - near closed set and X ∖ A ⊆ F r ∗ } = cl r ∗ ( X ∖ A )

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

Theorem 6. Let (O, F, _{∼ Br} , N _r  (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and A ⊆ X. x ∈ Cl r ∗ A if and only if N∩ A ≠ ∅  for all N ∈ N r ∗ ( x ) .

Proof. Let x ∈ Cl r ∗ A . Then x ∈ ∩ { F r ∗ : F r ∗ r-near closed set and A ⊆ F r ∗ } . Suppose that N∩ A = ∅ for N ∈ N r ∗ ( x ) . Since N ∈ N r ∗ ( x ) , then there exists a G r ∗ ∈ τ r ∗ such that x ∈ G ⊆ G r ∗ ⊆ N . Hence G r ∗ ∩ A = ∅ . If we choose F r ∗ = X ∖ G r ∗ , then A ⊆ F r ∗ and x ∉ F r ∗ . Consequently, x ∉ ∩ { F r ∗ : F r ∗ r-near closed set and A ⊆ F r ∗ } . This is contradiction.

Conversely, let N∩ A ≠ ∅ for all N ∈ N r ∗ ( x ) . Suppose that x ∉ Cl r ∗ A . Then x ∉ ∩ { F r ∗ : F r ∗ r-near closed set and A ⊆ F r ∗ } . Therefore, there exists a F r ∗ r-near closed set include A such that x ∉ F r ∗ . If we choose G r ∗ = X ∖ F r ∗ , G r ∗ ∈ τ r ∗ and G r ∗ ∈ N r ∗ ( x ) . Moreover, since A ⊆ F r ∗ , A ∩ G r ∗ = ∅ . This is contradiction.■

Definition 11. Let (O, F, _{∼ Br} , N _r  (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and A ⊆ X. The set of CoreA = { x ∈ X : x ∈ G ⊆ G r ∗ ⊆ A ve G ∈ τ} is called core of A.

Proposition 4. (O, F, _{∼ Br} , N _r  (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and A ⊆ X. Then the followings are true;

(1) CoreA ⊆ int r ∗ A

(2) int r ∗ A ⊆ intA

(3) clA ⊆ cl r ∗ A

Proof. (1)

x ∈ CoreA ⇒ x ∈ { x ∈ X : x ∈ G ⊆ G r ∗ ⊆ A ve G ∈ τ } ⇒ Then there exist G ∈ τ such that x ∈ G ⊆ G r ∗ ⊆ A .

⇒ Then there exist G r ∗ ∈ τ r ∗ such that x ∈ G r ∗ ⊆ A . ⇒ x ∈ int r ∗ A

(2) Let x ∈ int r ∗ A . Suppose that x ∉ intA. Then x ∉ ∪ {G ∈ τ : G ⊆ A}. Therefore G ⊈ A for all x ∈ G ∈ τ. Thus G r ∗ ⊈ A for all x∈ G r ∗ ∈ τ r ∗ . Consequently, x ∉ int r ∗ A . This is contradiction.

(3) Let x ∈ clA. Suppose that x ∉ cl r ∗ A . Then there exist N ∈ N r ∗ ( x ) such that N∩ A = ∅. Since N ∈ N r ∗ ( x ) , then there exist G ∈ τ such that x ∈ G ⊆ G r ∗ ⊆ N . Therefore, N ∈ N (x) and N∩ A = ∅. This is contradiction.■

3 Sequences

Definition 12. Let (O, F, _{∼ Br} , N _r  (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and (x _n ) be a sequence in X. The sequence (x _n ) convergence to x₀ and, denoted x n → r x 0 , if and only if each N ∈ N r ∗ ( x 0 ) , there is n 0 ∈ ℕ such that n ≥ n₀ implies x _n  ∈ N.

Example 4. Let X = {a, b, c, d, e, f}, B = {φ₁, φ₂, φ₃} be a set of functions representing object features and τ = {∅ , X, {b} , {a, b} , {c, d} , {b, c, d} , {a, b, c, d}}. Then the function φ _i defined as

Let us construct the equivalence classes for each r-combination. These equivalence classes are formed as

[a] _{φ1} = {a, c}, [b] _{φ1} = {b}, [d] _{φ1} = {d, e}

[a] _{φ2} = {a, c}, [b] _{φ2} = {b, d, e}

[a] _{φ3} = {a, c}, [b] _{φ3} = {b, e}, [d] _{φ3} = {d},

[a] _{φ1,φ2} = {a, c}, [b] _{φ1,φ2} = {b},[d] _{φ1,φ2} = {d, e}

[a] _{φ1,φ3} = {a, c}, [b] _{φ1,φ3} = {b}, [d] _{φ1,φ3} = {d}, [e] _{φ1,φ3} = {e}

[a] _{φ2,φ3} = {a, c}, [b] _{φ2,φ3} = {b, e}, [d] _{φ2,φ3} = {d}

[a] _{{φ1,φ2,φ3}} = {a, c}, [b] _{{φ1,φ2,φ3}} = {b}, [d] _{{φ1,φ2,φ3}} = {d}, [e] _{{φ1,φ2,φ3}} = {e}.

Then τ 1 ∗ = { ∅ , X , { b , d , e } } . Similiarly τ 2 ∗ = { ∅ , X , { b , e } , { a , b , c , e } , { a , c , d , e } } and τ 3 ∗ = { ∅ , X , { b } , { a , b , c } , { a , c , d } , { a , b , c , d } } and on the other hand

N 1 ∗ ( a ) = { X } , N 2 ∗ ( b ) = { X , { b , d , e } } , N 1 ∗ ( c ) = { X } , N 1 ∗ ( d ) = { X } , N 1 ∗ ( e ) = { X }

N 2 ∗ ( a ) = { { a , b , c , e } , X } , N 2 ∗ ( b ) = { U ⊆ X : { b , e } ⊆ U } , N 2 ∗ ( c ) = { X } , N 2 ∗ ( d ) = { { a , c , d , e } , X } , N 2 ∗ ( e ) = { X }

N 3 ∗ ( a ) = { U ⊆ X : { a , b , c } ⊆ U } , N 3 ∗ ( b ) = { U ⊆ X : { b } ⊆ U } , N 3 ∗ ( c ) = { U ⊆ X : { a , c , d } ⊆ U } , N 4 ∗ ( d ) = { U ⊆ X : { a , c , d } ⊆ U } , N 2 ∗ ( e ) = { X } .

The convergences of (x _n ) = (a, b, c, c, b, a, b, a, b, a, . . .) for the topologies above are

If we analyze the example above, it can be seen that there is a relationship between the convergence of (x _n ) on τ and on r-near topologies. Let us give this relationship with the following theorem.

Theorem 7. (O, F, _{∼ Br} , N _r  (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and (x _n ) be a sequence in X. Then followings are true;

(1) If x n → s x 0 , then x n → r x 0 , for all r ≤ s.

(2) If x _n  → x₀,  then x n → r x 0 , for all r ≤ |B|.

Proof. (1) Let r ≤ s and x n → s x 0 . Then there exist n 0 ∈ ℕ for all N ∈ N s ∗ ( x 0 ) such that x _n  ∈ N for all n ≥ n₀. By theorem 2, for all N ∈ N s ∗ ( x 0 ) , there exist n 1 ∈ ℕ such that (x _n ) ∈ N for all n ≥ n₁. Therefore x n → r x 0 .

(2) Let x _n  → x₀ and r ≤ |B|. Then there exist a n 0 ∈ ℕ for all N ∈ N (x₀) neighborhood such that x _n  ∈ N for all n ≥ n₀. Therefore, since all G open set, which contains x₀, is a neighborhood, there exist an n 1 ∈ ℕ integer such that x _n  ∈ G for all n ≥ n₁. Consequently, there exist an n 1 ∈ ℕ integer for all N ∈ N s ∗ ( x 0 ) r-near neighborhood such that x n ∈ G ⊆ G r ∗ ⊆ N for all n ≥ n₁.■

Conclusion 1. This article is the first study to investigate topological structures on near sets. The obtained results will be a reliable reference to the future researchs.

This work is supported by the Scientific Research Project Fund of Sivas Cumhuriyet University under the project number F-572.

Compliance with Ethical Standards:

Conflict of Interest: I (Serkan ATMACA) declare that I have no conflict of interest.

Ethical approval: This article does not contain any studies with human participants or animals performed by me.