In this manuscript, it is aimed to convert the topology on a set X which is on a nearness approximation space to new set families via indiscernibility relation. Then, if the open sets of the present topology are defined as the set of related elements, the set families, which have weakly related elements, will be obtained. Finally, the topological properties and concepts that these new families hold will be examined.
Even though the nearness concept is mentioned in earlier literature, near set concept is first given in the article “Near sets. Special theory about nearness of objects” by J. Peters in 2007. In this study, Peters construct the Nearness Approximation Space by using indiscernibility relation that he constructed via the functions representing object features. Following Peters article, many researchers applied this new set theory to different areas of mathematics ([1–10]) based on Nearness Approximation Space.
On the other hand, the nearness concept in mathematics was first encountered in the concept of neighborhood. Neighborhood is roughly defined as the set of the elements which are close to point with a certain distance. Outside of the conventional mathematics, it is also possible to refer to nearness by using the distance between two points in any metric space. In topological spaces, the concept of open sets is a natural way of defining the nearness of points. This nearness concept is the state of being the element of the same open set without being bound to a distance independent of the metric spaces. It can briefly be said that the nearness concept is a fundamental approximation of construction of topological spaces.
In this manuscript, it is aimed to convert the topology on a set X which is on a nearness approximation space to new set families via indiscernibility relation. Then, the set families, which have weakly related elements, will be obtained when the open sets of the present topology are defined as the set of related elements. Finally, the topological properties and concepts, these new families hold, will be examined.
Introduction
In this section, we give properties and some definitions of near sets defined by Peters [5, 6].
Object Description
Object Description
Symbol
Interpretation
Set of Real Numbers
O
Set of perceptual objects,
X
set of sample objects,
x
x ∈ O sample objects,
F
A set of functions representing object features,
B
B ⊆ F
Φ
Φ:O→ RL, object description,
L
is a description length,
i
i ≤ L
φi
φi: X → ℝ probe function, where φi ∈ B.
Φ (x)
Φ (x)=(φ1(x), φ2(x), ... , φL(x))
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 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 φi 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) , . . . , φL (x)), this will be a basis for the definition of the object.
∼ B = {(x, x′) ∈ O × O : Δφi = 0, ∀φi ∈ 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 φi ∈ B such that x∼ {φi} x′, Δφi = 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 ∼ Bbe 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
.
(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
.
(3) The boundary of A is denoted as BndBA and defined as
BndBA = B∗A ∖ B∗A = {x ∈ O : x ∈ B∗A and x ∉ B∗A}.
Nearness Approximation Space
Nearness Approximation Space Symbols
Symbol
Interpretation
B
B ⊆ F
Br
r ≤ | B | probe functions in B,
∼ Br
Br Indiscernibility relation defined using Br,
[x]Br
[x]Br = {x′ ∈ X: x ∼ Brx′}, equivalence class,
O/∼ Br
O/∼ Br ={ [x]Br: x∈ O}, quotient set,
Nr(B)
Nr(B) ={ O/∼ Br: Br ⊆ B}, set of all O/∼ Br partitions,
Nr(B) ∗ A
Nr(B) ∗ A= ∪ [x]Br ⊆ A [x]Br, lower approximation,
Nr(B) ∗ A
Nr(B) ∗ A = ∪ [x]Br ∩ A ≠ ∅ [x]Br, upper approximation
Indiscernibility relation can be defined for each subset Br, such that Br ⊆ B ⊆ F and |Br| = r. Let us denote that relation with ∼ Br. ∼ Brcan form different decomposition for each r over O. Here, ∼ Brseparate O to [x] Br nearness classes and O/∼ Br = {[x] Br : x ∈ O} set is quotient set. Consequently, Nr (B) = {O/∼ Br : Br ⊆ B} set of all O/∼ Brpartitions 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, ∼ Br, Nr (B)) is called Nearness Approximation Space.
Definition 6. [5] Let (O, F, ∼ Br, Nr (B)) be a Nearness Approximation Space and A ⊆ O.
(1) The union of [x] Br ∈ O/∼ Br elements, whose are subset of A, is called Br lower Approximation of A and defined as
.
(2) The union of [x] Br ∈ O/∼ Br elements, whose intersection with A is non-empty, is called Br upper Approximation of A and defined as
.
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), Nr (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 Nr (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.
r-Near topological spaces
Definition 7. Let (O, F, ∼ Br, Nr (B)) be a Nearness Approximation Space, X ⊆ O and (X, τ) be a topological space. The family is called r-near topology which generated by (O, F, ∼ Br, Nr (B)). The elements Nr (B) ∗ (G) are called r-near open sets. Complement of r-near open sets are called r-near closed sets.
Throughout the paper, Nr (B) ∗ (G) r-near open sets are denoted as for simplicity unless specified otherwise. The family of all r-near closed sets is denoted as .
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 φi defined as
a
b
c
d
e
f
φ1
0
1
0
1
0
1
φ2
0
1
2
0
1
1
φ3
0
1
2
3
0
1.
Let us construct the equivalence classes for each r-combination. These equivalence classes are formed as
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 r-near topology. Where
N1(B)∗ (∅) = ∅
N2(B)∗ (∅) = ∅
N3(B)∗ (∅) = ∅
N1(B)∗ (X) = X
N2(B)∗ (X) = X
N3(B)∗ (X) = X
N1(B)∗ ({c}) = {c}
N2(B)∗ ({c}) = {c}
N3(B)∗ ({c}) = {c}
N1(B)∗ ({a, b, c}) = X
N2(B)∗ ({a, b, c}) = X
N3(B)∗ ({a, b, c}) = X
N1(B)∗ ({c, d, e, f}) = X
N2(B)∗ ({c, d, e, f}) = X
N3(B)∗ ({c, d, e, f}) = X
and . Similiarly and .
As seen in the Example 2, even-though and families are topology, family is not a topology. Let us show which of the terms of being topology that 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, Nr (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 and .
■
Theorem 2.Let (O, F, ∼ Br, Nr (B)) be a Nearness Approximation Space, X ⊆ O and (X, τ) be a topological space. If , for all i ∈ I, then where I is an arbitrary index set.
Proof. Let , for all i ∈ I. Then
. On the other hand,
. Since τ is a topology, , that is .■
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, Nr (B)) be a Nearness Approximation Space, X ⊆ O and (X, τ) be a topological space. The families are supra topology on X.
Theorem 3. Let (O, F, ∼ Br, Nr (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 .
(2) If Br ⊆ Bs, then [x] Bs ⊆ [x] Br.
(3) , for all G ∈ τ.
(4) , for all G ∈ τ.
(5) If G ∈ τ and x0 ∈ G, then is a neighborhood of x0.
Proof. (1) Let G ⊆ H. Then .
(2)
(3)
(4)
(5) Obvious from (3).■
Definition 9. Let (O, F, ∼ Br, Nr (B)) be a Nearness Approximation Space, X ⊆ O and (X, τ) be a topological space. If there exists a such that , then N is called r-near neighborhood of x. If Nr-near open set, then N is called r-near open neighborhood.
The family of all r-near neighborhood of x is denoted and the family of all r-near open neighborhood is denoted as .
Example 2. If we consider Nearness Approxi-mation Space in Example 2 and . Then and .
Proposition 1. Let (O, F, ∼ Br, Nr (B)) be a Nearness Approximation Space, X ⊆ O and (X, τ) be a topological space. Then followings are true;
(1) If , then x ∈ N.
(2) If and N1 ⊆ N2, then .
(3) If N1,, then .
Proof. (1) Let . Then there exists a such that . Therefore, x ∈ N.
(2) Let and N1 ⊆ N2. Then there exists a such that . Therefore, .
(3) Let N1, . Then there exist , such that and . Since τ is a topology, G ∩ H ∈ τ, and . Consequently, .■
Proposition 2.Let (O, F, ∼ Br, Nr (B)) be a Nearness Approximation Space, X ⊆ O and (X, τ) be a topological space. If U neighborhood of x, then Nr (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 , . This completes the proof.■
Theorem 4.Let (O, F, ∼ Br, Nr (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and r ≤ s ≤ |B|. If for all x0 ∈ X, then .
Proof. Let . Then there exists a such that . Since by Theorem 2 (4), . This completes the proof.■
Definition 10. Let (O, F, ∼ Br, Nr (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and A ⊆ X.
(1) The set is a r-near closed set and is called r-near closure of A .
(2) The set is a r-near open set and is called r-near interior of A .
Example 3. Let us consider the Nearness Approximation Space and {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 and the closure of A is .
Theorem 5. Let (O, F, ∼ Br, Nr (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 .
(2) If A ⊆ B, then .
Proof. (1)
(2)
■
Proposition 3. Let (O, F, ∼ Br, Nr (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and A ⊆ X. Then the followings are true;
(1)
(2)
Proof. (1)
(2) The proof is similar to proof (1).■
Theorem 6.Let (O, F, ∼ Br, Nr (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and A ⊆ X. if and only if N∩ A ≠ ∅ for all .
Proof. Let . Then r-near closed set and . Suppose that N∩ A = ∅ for . Since , then there exists a such that . Hence . If we choose , then and . Consequently, r-near closed set and . This is contradiction.
Conversely, let N∩ A ≠ ∅ for all . Suppose that . Then r-near closed set and . Therefore, there exists a r-near closed set include A such that . If we choose , and . Moreover, since , . This is contradiction.■
Definition 11. Let (O, F, ∼ Br, Nr (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and A ⊆ X. The set of ve G ∈ τ} is called core of A.
Proposition 4. (O, F, ∼ Br, Nr (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and A ⊆ X. Then the followings are true;
(1)
(2)
(3)
Proof. (1)
(2) Let . Suppose that x ∉ intA. Then x ∉ ∪ {G ∈ τ : G ⊆ A}. Therefore G ⊈ A for all x ∈ G ∈ τ. Thus for all x∈ . Consequently, . This is contradiction.
(3) Let x ∈ clA. Suppose that . Then there exist such that N∩ A = ∅. Since , then there exist G ∈ τ such that . Therefore, N ∈ N (x) and N∩ A = ∅. This is contradiction.■
Sequences
Definition 12. Let (O, F, ∼ Br, Nr (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and (xn) be a sequence in X. The sequence (xn) convergence to x0 and, denoted , if and only if each , there is such that n ≥ n0 implies xn ∈ 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 φi defined as
a
b
c
d
e
φ1
0
1
0
2
2
φ2
0
1
0
1
1
φ3
0
2
0
1
2
Let us construct the equivalence classes for each r-combination. These equivalence classes are formed as
The convergences of (xn) = (a, b, c, c, b, a, b, a, b, a, . . .) for the topologies above are
xn → a
xna
xna
xna
xn ↛ b
xnb
xnb
xnb
xn ↛ c
xnc
xnc
xnc
xn ↛ d
xnd
xnd
xnd
xn → e
xne
xne
xne.
If we analyze the example above, it can be seen that there is a relationship between the convergence of (xn) on τ and on r-near topologies. Let us give this relationship with the following theorem.
Theorem 7. (O, F, ∼ Br, Nr (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and (xn) be a sequence in X. Then followings are true;
(1) If , then , for all r ≤ s.
(2) If xn → x0, then , for all r ≤ |B|.
Proof. (1) Let r ≤ s and . Then there exist for all such that xn ∈ N for all n ≥ n0. By theorem 2, for all there exist such that (xn) ∈ N for all n ≥ n1. Therefore .
(2) Let xn → x0 and r ≤ |B|. Then there exist a for all N ∈ N (x0) neighborhood such that xn ∈ N for all n ≥ n0. Therefore, since all G open set, which contains x0, is a neighborhood, there exist an integer such that xn ∈ G for all n ≥ n1. Consequently, there exist an integer for all r-near neighborhood such that 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.
References
1.
HenryC., Near Sets: Theory andApplications (Ph.D. Thesis (supervisor J.F. Peters) Department of Electrical & Computer Engineering, University of Manitoba, (2010).
2.
İnanE. and ÖztürkM.A., Erratum and notes for near groups on nearness approximation spaces, Hacettepe Journal of Mathematics and Statistics43(2) (2014), 279–281.
3.
İnanE. and ÖztürkM.A., Near groups on nearness approximation spaces, Hacettepe Journal of Mathematics and Statistics41(4) (2012), 545–558.
4.
İnanE. and ÖztürkM.A., Near semigroups on nearness approximation spaces, Annals of Fuzzy Mathematics and Informatics10(2) (2015), 287–297.
5.
PetersJ.F., Near sets, General theory about nearness of objects, Applied Mathematical Sciences1(53–56) (2007), 2609–2629.
6.
PetersJ.F., Near sets, special theory about nearness of objects, Fundamenta Informaticae75(1–4) (2007), 407–433.
7.
PetersJ.F., Sufficiently near sets of neighbourhoods, Rough Sets and Knowledge Technology (2011), LNCS 6954, Springer, Berlin, 17–24.
8.
PetersJ.F. and NaimpallyS., Approach spaces for near filters, General Mathematical Notes2(1) (2011), 159–164.
9.
PetersJ.F. and TiwariS., Approach merotopies and near filters, General Mathematical Notes3(1) (2011), 1–15.
10.
WolskiM., Perception and classification, A note on near sets and rough sets, Fundamenta Informaticae101 (2010), 143–155.
11.
MashhourA.S., et al., On supra topological spaces, Indian J Pure Appl Math14(4) (1983), 502–510.