Download PDF - International Journal of Mathematical Archive

International Journal of Mathematical Archive-4(1), 2013, 182-187
Available online through www.ijma.info ISSN 2229 – 5046
SOME TOPOLOGICAL APPLICATIONS ON ROUGH SETS
A. M. Koze1 & A. I. EL-Maghrabi2-3
1
Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt
Department of Mathematics, Faculty of Science, Taibah University, AL-Madinah AL-Munawarh,
P.O. Box, 344, AL-Madinah, K.S.A.
3
Department of Mathematics, Faculty of Science, Kafr EL-Sheikh University, Kafr EL-Sheikh, Egypt
2
(Received on: 12-12-12; Revised & Accepted on: 23-01-13)
ABSTRACT
The aim of this paper is devoted to give and study the notion of γ- rough sets and γ- rough pairs. Also, some of their
basic characterizations and properties are investigated. Further, the relation between this topology and some
separation axioms say: γ-T0 and γ-T1 which well known above are discussed.
AMS Subject Classification: 04A, 03B, 54A.
Keywords and Phrases: γ–rough set, γ-rough pair, γ g-closed, γ-dense, γ –co-dense sets, γ-symmetric and Alexandroff
spaces,
1. INTRODUCTION
In 1982, Pawlak [19] introduced the concept of rough sets on an approximation space. Wiweger [22], in 1989
introduced the study of topological rough sets on a quasi- discrete space. Abd EL-Monsef et.al [1] discussed the
notions of s-exact and s-rough sets for a general approximation space (X, R). γ -open [10] (= b-open [2]) sets were
introduced by EL-Atik and Andrijevic. C. E. Aull [3] (or Levine [16]) introduced the notion of g-closed sets. The
purpose of this paper is to give and investigate the concepts of γ- rough sets and γ- rough pairs. Also, some of their
basic characterizations and properties are investigated. Further, the relation between this topology and some separation
axioms say: γ-T0 and γ-T1 which well known above are discussed.
2. PRELIMINARIES
Throughout the present paper (X, R) is a general approximation space, R is a binary relation on a universal set X and τR
is the topology on X associated with R. Let A be a subset of a space X. We denote the closure of A, the interior of A
and the complement of A by cl(A), int(A) and Ac respectively. A subset A of a space X is said to be semi-open [15]
(resp. preopen[18], γ-open [10]) if A⊆cl(int(A)) (resp. A⊆ 𝑖𝑖𝑖𝑖𝑖𝑖�𝑐𝑐𝑐𝑐(𝐴𝐴)�, 𝐴𝐴 ⊆ 𝑐𝑐𝑐𝑐(𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)) ∪ 𝑖𝑖𝑖𝑖𝑖𝑖(𝑐𝑐𝑐𝑐(𝐴𝐴)). The complement
of a semi-open (resp. preopen, γ –open) set is semi-closed [8] (resp. preclosed, γ -closed). The intersection of all semiclosed (resp. preclosed, γ -closed) sets containing A is called the semi-closure (resp. preclosure, γ - closure) of A and is
denoted by scl(A) (resp.pcl(A), γ-cl(A)). The union of all semi-open (resp. preopen, γ –open) sets contained in A is
called the semi-interior ( resp. pre-interior, γ - interior ) of A and is denoted by s-int(A) ( resp. p-int(A), γ - int(A)).The
family of all γ –open (resp. γ –closed) sets will be denoted by γO(X ) (resp. γC(X)).
Definition 2.1. A subset A of a space (X, τ) is called:
(1) a generalized closed (briefly, g-closed) [3,16] set if cl(A) ⊆ U whenever A⊆ U and U is open in (X,τ),
(2) a semi-generalized closed ( briefly, sg-closed) [4] set if scl(A) ⊆ U whenever A⊆ U and U is semi-open in (X,τ),
(3) a γ -generalized closed ( briefly, γ g-closed) [12] set if γ -cl(A) ⊆ U whenever A⊆ U and U is open in (X,τ).
Definition 2.2[20]. Let U be a finite non-empty set called the Universe and R be a binary relation on U. By xR, we
mean the set of all y such that xRy. If we assume that R is reflexive and symmetric, then R is called a tolerance relation
on U.
Definition 2.3[11]. A space (X, τ) is said to be:
(1) a γ-T0 space if for every x, y∈ 𝑋𝑋, 𝑥𝑥 ≠ 𝑦𝑦, there exists a γ-open set U of X such that either x∈ 𝑈𝑈, y∉ U or x ∉U, y ∈U,
(2) a γ-T1 space if for every x,y∈ 𝑋𝑋, 𝑥𝑥 ≠ 𝑦𝑦, there exist two γ-open sets U,V of X such that x∈ 𝑈𝑈, y∉ U and x ∉V, y ∈V.
3
Corresponding author: A. I. EL-Maghrabi2-3
Department of Mathematics, Faculty of Science, Kafr EL-Sheikh University, Kafr EL-Sheikh, Egypt
International Journal of Mathematical Archive- 4(1), Jan. – 2013
182
1
2-3
A. M. Koze & A. I. EL-Maghrabi / Some topological applications on rough sets/IJMA- 4(1), Jan.-2013.
Definition 2.4[22]. A topology σ on a se U is called quasi- discrete if σ is equal to the set of all closed sets in (U, σ).
Definition 2.5[21]. A set A is called a topological Boolean algebra with respect to the operations ∩,∪, 𝑐𝑐 𝑎𝑎𝑎𝑎𝑎𝑎 𝑖𝑖𝑖𝑖𝑖𝑖 if it
satisfies the conditions:
(1) A is a Boolean algebra with respect to ∩,∪ 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐,
(2) if x ∈A, then int(x) ∈ A,
(3) if x ∈A, then int(x) ≤ A,
(4) if x ∈A, then int( int(x)) = 𝑖𝑖𝑖𝑖𝑖𝑖(𝑥𝑥),
(5) if x and y are in A, then int(x∩ 𝑦𝑦)= 𝑖𝑖𝑖𝑖𝑖𝑖(𝑥𝑥) ∩ 𝑖𝑖𝑖𝑖𝑖𝑖(𝑦𝑦),
(6) int(1)= 1.
Definition 2.6[7]. A topological Boolean algebra A and the rough equality relation (≈) is called an approximation
algebra.
Definition 2.7. Let A be a topological Boolean algebra and x,y are elements of A. Then:
(1) The elements are roughly equal (x ≈y)[5] if int(x) =I nt(y) and cl(x) = cl(y),
(2) Equivalence classes of the rough equality relation are called rough sets in (A, ≈) [6,22],
(3) In an approximation algebra (A, ≈), rough sets [x] ≈ which satisfy the condition int(x) = cl(x) are called exact sets
[7].
Definition 2.8[1]. If X is a universal set and R is a binary on X, then the pair (X, R) is called a general approximation
space.
Definition 2.9[1]. If (X, R) is called a general approximation space, then the topology τ R associated with (X, R) is the
topology generated by {xR: x ∈X}.
Lemma 2.1[14]. (X, τR) is an Alexandroff space.
Definition 2.10[1]. Let (X, τR) be the space associated with (X, R) and A be a subset of X. Then A is said to be s-exact
if int(cl(A))=cl(int(A)).
Theorem 2.1[11]. Let (X, τ) be a space. Then the following statements are equivalent:
(1) (X, τ) is a γ- T1 space,
(2) For any point x∈ X, the singleton set {x} is γ- closed.
3. Some properties of γ -rough sets
In this article, we give some properties on γ -rough sets. Also, we investigate some relations between τR and some
separation axioms.
Definition 3.1. Let (X, τR) be the space associated with (X, R) and A be a subset of X. Then A is called:
(1) γ-exact if 𝛾𝛾 −int (A) = γ- cl (A),
(2) γ-rough if 𝛾𝛾 −int (A)) ≠ γ- cl (A)
Example 3.1. Let X={x, y, z, u} with a binary relation R= {(x, y), (x, z), (y, z), (y, u), (z, u)}. Then a topology τR
associated with (X, R) is τR = {𝜑𝜑, {𝑧𝑧}, {𝑢𝑢}, {𝑧𝑧, 𝑢𝑢} , {𝑦𝑦, 𝑧𝑧}, {𝑦𝑦, 𝑧𝑧, 𝑢𝑢}, 𝑋𝑋}. Further, the subsets {u}, {y, z}, {x, u} are
γ-exact but {z}, {z, u}, {y, z, u} are γ-rough.
Remark 3.1. Every rough set [19] is γ-rough but the converse is not true as is shown by the following example.
Example 3.2. In Example 3.1, the subsets {z, u}, {y, z, u} are γ-rough sets but not rough sets.
Example 3.3. Let X= {a, b, c} with binary operation R= {(a, b),(a, c)}. Then a topology τR associated with (X, R) is
τR ={𝜑𝜑, {𝑏𝑏, 𝑐𝑐}, 𝑋𝑋}. Further, the subsets {b}, {a, b},{a, c} are γ-exact but not s-exact and the subsets {a}, {b, c} are γrough.
Lemma 3.1. For any topology τR associated with a general approximation space (X, R) and for all a, b in X, the
condition a∈ 𝛾𝛾-cl({b}) and b ∈ 𝛾𝛾-cl({a}) implies 𝛾𝛾-cl({a}) =𝛾𝛾-cl({b}).
Proof. Since 𝛾𝛾-cl({b}) is a γ –closed set containing a, while 𝛾𝛾-cl({a}) is the smallest γ – closed set containing a, hence
𝛾𝛾-cl({a}) ⊆ 𝛾𝛾-cl({b})
© 2013, IJMA. All Rights Reserved
(1)
183
1
2-3
A. M. Koze & A. I. EL-Maghrabi / Some topological applications on rough sets/IJMA- 4(1), Jan.-2013.
The converse is similar, we have 𝛾𝛾-cl({b}) ⊆ 𝛾𝛾-cl({a})
Then from (1), (2), we have 𝛾𝛾-cl({a}) =𝛾𝛾-cl({b}).
(2)
Lemma 3.2. If τR is a topology associated with a general approximation space (X,R) and every γ –open set of X is
γ –closed, then b∈ 𝛾𝛾-cl({a}) implies a ∈ 𝛾𝛾-cl({b}), for all a, b ∈ X.
Proof. If a ∉ 𝛾𝛾-cl({b}), then there exists a γ –open set H containing a such that H ∩{b}= φ implies that {b}⊆ Hc which
is γ –closed and also is a γ –open set does not containing a, then Hc ∩{a}= φ which implies that b ∉ 𝛾𝛾-cl({a}).
Definition 3.2. A space (X, τ) is called γ–symmetric if , for x, y∈ X, then x ∈ 𝛾𝛾-cl({y}) implies that y ∈ 𝛾𝛾-cl({x}).
Remark 3.2. According to Definition 2. [20] and Lemma 3.2, we can say that a γ–symmetric space is equivalent to
(X, τR) in which every γ –open set of X is γ –closed.
Proposition 3.1. Let τR be a topology associated with a general approximation space (X, R) and every γ –open set of X
is γ –closed. Then the family of sets 𝛾𝛾-cl({a}), a∈ X is a partition of the set X.
Proof. If a,b,c ∈ X and c ∈ 𝛾𝛾-cl({a}) ∩ 𝛾𝛾-cl({b}) implies that c ∈ 𝛾𝛾-cl({a}) and c ∈ ∩ 𝛾𝛾-cl({b}), then by Lemma 3.1,
𝛾𝛾-cl({a}) =𝛾𝛾-cl({c}) and 𝛾𝛾-cl({b}) =𝛾𝛾-cl({c}). Hence 𝛾𝛾-cl({a}) =𝛾𝛾-cl({b}) =𝛾𝛾-cl({c}). Therefore, either 𝛾𝛾-cl({a})=
𝛾𝛾-cl({b}) or 𝛾𝛾-cl({a}) ∩ 𝛾𝛾-cl({b})=φ. The proof is complete.
Definition 3.3. A subset A of a space X is called generalized 𝛾𝛾 –closed (briefly, g 𝛾𝛾 -closed) if 𝛾𝛾-cl({A})⊆ 𝐻𝐻 whenever
A ⊆ 𝐻𝐻 and H is 𝛾𝛾 –open in (X,τ).
Remark 3.3. The concepts of 𝛾𝛾 –closed [10] and g 𝛾𝛾 −closed sets are coincident.
Example 3.4. If X={u, v, w, z} and τ={φ, {w, z}, X}, then a subset A={u, v, z} is 𝛾𝛾 –closed and g 𝛾𝛾 −closed.
Remark 3.4. Every gγ-closed set is γg-closed [12]. If X= {a, b, c}, τ ={φ, {a},{b}, {a, b},X} and A={a}, then A is
γg-closed but it is not gγ-closed.
Corollary 3.1. If B is gγ-closed which also open and V is 𝛾𝛾 –closed, then B∩ 𝑉𝑉 is gγ-closed in B.
Proof. Since 𝛾𝛾𝛾𝛾𝛾𝛾(𝐵𝐵) = 𝐵𝐵, if B is γ -closed [10, Theorem 1.2], then B∩ 𝑉𝑉 is γ-closed in X and this implies that B∩ 𝑉𝑉 is
γ-closed in B. Hence B∩ 𝑉𝑉 is gγ-closed in B.
Theorem 3.1. For a topological space (X, τ), then the following are equivalent:
(1) every gγ-closed set is γ-closed,
(2) for every singleton {x} of X, {x} is γ-open or γ-closed.
Proof. By [17, Definition 2.10], it is shown that a subset B is gγ-closed if and only if A is (𝛾𝛾𝛾𝛾(𝑋𝑋), 𝛾𝛾𝛾𝛾(𝑋𝑋)) −
𝑔𝑔 −closed and by [17, Definition 2.19], a space X is (𝛾𝛾𝛾𝛾(𝑋𝑋), 𝛾𝛾𝛾𝛾(𝑋𝑋)) − 𝑇𝑇1/2 if and only if every gγ-closed set is
γ-closed. Since the intersection of arbitrary γ-closed sets is γ-closed, hence a property (C) in [17, Lemma 2.20(iii)] is
satisfied. Therefore by [17, Lemma 2.20(iii)], (1) and (2) are equivalent.
Theorem 3.2. For a topological space (X, τ), every singleton is γ-open or γ-closed.
Proof. By [13, Lemma 2], every singleton {x} is nowhere dense or preopen. For any point x ∈ 𝑋𝑋, we suppose the
following two cases:
Case 1: {x} is nowhere dense. Since int-cl({x})=φ, we have φ = int-cl({x}) ∩ cl-int({x})⊆ {𝑥𝑥} and so, {x} is γ-closed.
Case 2: {x} is preopen. Since {x}⊆ int-cl({x}), we have {x}⊆ int-cl({x}) ∪ cl-int({x}) and so, {x} is γ-open .
Therefore every singleton of a space (X, τ) is γ-open or γ-closed.
Theorem 3.3. For a space (X, τR ), then the following statements are equivalent:
(1) every γ-open set of X is γ-closed,
(2) {x} is a gγ-closed set , for each x in X.
Proof. (1) → (2). Assume that {x}⊆ H which is γ-open but γcl({x}) ⊈H. Then γcl({x}) ∩Hc≠ 𝜑𝜑, take
y ∈γcl({x}) ∩Hc. Hence x ∈γcl({y}) ⊆Hc and x ∉H which is a contradiction.
© 2013, IJMA. All Rights Reserved
184
1
2-3
A. M. Koze & A. I. EL-Maghrabi / Some topological applications on rough sets/IJMA- 4(1), Jan.-2013.
(2)→(1). Suppose that x ∈γcl({y}) but y ∉γcl({x}). Then {y} ⊆X\ γcl({x}) and thus γcl({y})⊆X\ γcl({x}) and hence
x∈ X\ γcl({x}) which is a contradiction.
Corollary 3.2. If (X, τR ) is a γ –T1 space, then every γ-open set of X is γ-closed.
Proof. In a γ –T1 space, the singleton sets are γ-closed and therefore gγ-closed. Hence by Theorem 3.3, the space (X,
τR) is a topology associated with a general approximation space (X, R) and every γ –open set of X is γ –closed.
Corollary 3.3. For a space (X, τR), the following are equivalent:
(1) (X, τR ) is γ –T0 and every γ-open set of X is γ-closed,
(2) (X, τR ) is a γ –T1 space.
Proof. (1)→(2). Let x ≠y and ( X,τR ) be a γ –T0 space. We may assume that x∈ H⊆ X\({y}), for some γ-open set H.
Hence x∈ 𝛾𝛾cl({y}) and then y∉ 𝛾𝛾cl({x}). Therefore there exists a γ-open set U such that y∈ U⊆ X\({x}). Then (X, τR)
is γ –T1.
(2)→ (1). Obvious by Corollary 3.1 and Theorem 3.3.
Theorem 3.4. Let (X, τ R ) be a space in which every γ-open set of X is γ-closed. Then the following are equivalent:
(1) (X, τR) is a γ –T0 space,
(2) (X, τR) is a γ –T1 space.
Proof. Obvious from Corollary 3.3.
4. On γ- rough classes.
The notions of γ-closure and γ-interior are applied to construct a class of γ-roughness wider than the class of
topological rough sets.
Definition 4.1. If τR is a topology associated with a general approximation space (X, R) and A is a subset of X, then a
γ-rough class defined by A is determined by the pair (γ-int(A), γ-cl(A)),
where (γ-int(A), γ-cl(A))={B⊆ 𝑋𝑋: (𝛄𝛄 − int(A) ⊆ 𝐵𝐵 ⊆ 𝛄𝛄 − cl(A))} .
It is easy to see that each topological rough class is a topological γ –rough class.
Definition 4.2. If (X, τ R) is a topological space associated with a general approximation space (X, R), then a subset B
of X is said to be γ-dense (resp. γ-co-dense) if 𝛄𝛄 − cl(B) = 𝑋𝑋(𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟. 𝛄𝛄 − int(B) = 𝜑𝜑).
By γ –rough pair in a topological space (X τ R), we shall mean any pair (K, W), where K and W are subsets of τR
satisfying the following four conditions:
(γ1) K is γ-open,
(γ2) W is γ-closed,
(γ3) K ⊆ 𝑊𝑊,
(γ4) The set W\ γ -cl(K) contains a subset C which is at the same time γ - dense in W\ γ -cl(K) and γ –co-dense in X.
Condition (γ4) means that there is a set C such that:
(γ/1) γ− 𝑖𝑖𝑖𝑖𝑖𝑖 (𝐵𝐵) = 𝜑𝜑,
(γ/2) C ⊆ W\ 𝛄𝛄 − cl(K) ,
(γ/3) W\ γ -cl(K) ⊆ 𝛄𝛄 − cl(C).
Lemma 4.1. For any subset A of X, the pair (γ-int(A), γ-cl(A)) is a γ –rough pair in (X, τ R ) in which every γ-open set
of X is γ-closed.
Proof. Let K= γ -int(A), W= γ -cl(A). Then the conditions (γ1) → (γ3) are satisfied. To prove (γ4). Define C=A\ γ-cl(K).
(γ/1) If H is a γ –open set contained in C( =H⊆C), hence H⊆A implies that H ∩K= 𝜑𝜑 which is a contradiction. So, H
does not contain in A. Hence, C= 𝜑𝜑 implies that γ− 𝑖𝑖𝑖𝑖𝑖𝑖 (𝐶𝐶) = 𝜑𝜑.
(γ/2) Since C=A\ γ -cl(K), but A⊆ γ-cl(A), hence A⊆W, then C⊆W\ γ -cl(K).
(γ/3) Let x ∈ W\ γ -cl(K), where W= γ-cl(A) means that x∈ A or x ∉A.
If x∈ A, then for each γ –open set H contains a such that H ∩A≠ 𝜑𝜑 implies that H ∩C≠ 𝜑𝜑 and hence x ∈ γ -cl(C), then
W\ γ -cl(K) ⊆ γ-cl(C). OR, if x ∉A, then there is a γ –open set H contains x. Since H\ γ -cl(K)=H∩ [ γ -cl(K)]c is
© 2013, IJMA. All Rights Reserved
185
1
2-3
A. M. Koze & A. I. EL-Maghrabi / Some topological applications on rough sets/IJMA- 4(1), Jan.-2013.
γ –open contains x and x ∈ γ -cl(A), then there is y∈ A such that y ∈ H\ γ -cl(K), we have y∈ A∩ (H\ γ -cl(K)⊆ C,
hence y∈ H ∩C. Therefore H ∩C≠ 𝜑𝜑 and hence x ∈ γ -cl(C), then W\ γ -cl(K) ⊆ γ-cl(C). The proof is complete.
Lemma 4.2. For any γ –rough pair (K,W) in (X, τ R ) in which every γ-open set of X is γ-closed, then there is a subset
A of X such that K= γ -int(A) and W= γ-cl(A).
Proof. Let (K, W) be any γ –rough pair and let C be a set satisfying (γ/1) → (γ/3). Define A= K ∪C, then K ⊆A implies
that by using (γ1), K ⊆ γ -int(A). If H is a γ-open set contained in A, then H\ γ -cl(K)= H∩ [ γ -cl(K)]c is γ-open
contained in A. Since H⊆ A= K ∪C, but K⊆ γ-cl(K), then H⊆ γ-cl(K) ∪C and hence H\ γ -cl(K) ⊆ [(γ-cl(K) ∪C)\ γcl(K)]=C. Therefore H\ γ -cl(K) is γ-open contained in C which means that by using (γ/1), H⊆ γ-cl(K). Since H⊆ K
∪C, it follows that by using (γ/2) , H⊆ K and this proves that K= γ -int(A).
Another part of the proof. Since A= K ∪C, then γ-cl(A).= γ-cl(K ∪C)⊇ γ-cl(K) ∪ γ-cl(C) ⊇ 𝑊𝑊 ∪ γ-cl(C) ⊇ 𝑊𝑊.
Hence, γ-cl(A)⊇ 𝑊𝑊
(1)
The other side. Since W= γ-cl(K) ∪ (W\ γ -cl(K)), but W= γ-cl(W) = γ-cl[γ-cl(K) ∪ (W\ γ -cl(K)) ⊇
γ -cl(K) ∪ 𝛄𝛄 − cl(W\ γ -cl(K)) )⊇ γ-cl(K) ∪ γ-cl( C) ⊇ 𝛄𝛄 − cl(K ∪ C)= γ-cl(A) ( by using (γ/2)). Then by using (γ2),
γ-cl(A) ⊆ 𝑊𝑊
(2)
Then from (1),(2), we have W= γ-cl(A). End the proof.
If (X, τ R) is the space associated with a general approximation space (X, R) and A⊆ X, then τ*R is the relative topology
on A with respect to τ R [2].
Definition 4.3. If τ*R is a topological γ –rough class in (X, τ R ), we define f(τ*R )=(γ-int(A), γ-cl(A)), where A∈ τ*R .
Theorem 4.1. For any topological space (X, τ R) associated with a general approximation space (X,R), the function
f(τ*R )=(γ-int(A), γ-cl(A)), where A∈ τ*R is a bijection from the set of all γ –rough classes in (A, τ*R ) onto the set of
all γ –rough pairs in (X, τ R ) .
Proof. (1) f is injection: If f(A1)=f(A2) implies that (γ-int(A1), γ-cl(A1)) = (γ-int(A2), γ-cl(A2) implies that γ-int(A1) =
γ-int(A2), γ-cl(A1)= γ-cl(A2). Hence A1≈A2.
(3) F is surjective: for any γ –rough pair (γ-int(A), γ-cl(A)) in (X, τ R ), then there exists A∈ τ*R such that f(A)=
(γ-int(A), γ-cl(A)). The proof is complete.
CONCLUSION
The initiation of the study of rough sets on an approximation space by Pawlak [19] in 1982. A large number of authors
have turned their attention to the generalization of approximation spaces. In 1989, Wiweger [22] introduced the study
of topological rough sets on a quasi-discrete space. Alexandroff space plays a significant role in the digital topology
researches. The properties of this space is too rich compared with the quasi-discrete space. Abd EL-Monsef et.al[1] in
2003, introduced and studied the concepts of s-exact and s-rough sets for a general approximation space (X,R). The
topology associated with Pawlak,s approximation space is a quasi-discrete topology. Since R in this case is an
equivalence relation. While the topology τR associated with a general approximation space (X, R) is a topological space
having the property that arbitrary intersection of open sets is open [14].
In this paper, we define the concept of 𝛄𝛄 − 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟ℎ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 and γ − rough pairs. There is no a unique way to define these
concepts and either approximations or rough membership. Also, we try to find some properties and characterizations
between a topology τR associated with a general approximation space (X,R) and some separation axioms such as: γ-T0 ,
γ-T1 spaces which well known from above. To understand better the relationship between them some results are
introduced.
ACKNOWLEDGEMENT
We are thankful to Professor H. Maki (JAPAN) for many comments and suggestions Theorems 3.1, 3.2.
REFERENCES
[1] M. E. Abd EL-Monsef; A. M. Kozae and A. I. EL-Maghrabi, Some semi-topological applications on rough sets, J.
Egypt. Math. Soc., 12(1) (2004), 45-53.
[2] D. Andrijevic, On b-open sets, Mate. Bechnk, 48(1996), 59-64.
© 2013, IJMA. All Rights Reserved
186
1
2-3
A. M. Koze & A. I. EL-Maghrabi / Some topological applications on rough sets/IJMA- 4(1), Jan.-2013.
[3] C. E. Aull, Paracompact and countably paracompact subsets, General Topology and its relation to modern Analysis
and Algebra, Proc. Kanpur Topological Con., (1968), 49-53.
[4] P. Bhattacharyya and B.K. Lahiri, Semi-generalized closed sets in topology, Indian J. Math., 29(1987), 375-382.
[5] Z. Bonikowski, A certain conception of the calculus of rough sets, Notre Dame Journal of Formal Logic, 33(3)
(1992), 412-421.
[6] M. Chuchro, A certain conception of rough sets in topological Boolean Algebra, Bulletin of the section of Logic,
22(1) (1993), 9-12.
[7] M. Chuchro, On rough sets in topological Boolean Algebras, Collection: Rough sets , Fuzzy sets and Knowledge
discovery (Banff, AB, 1993), 157-160.
[8] S. G. Crossely and S.K. Hildebrand, Semi-closure, Texas J. Sci., 22(1971), 99-112.
[9] J. Dontchev and M. Przemski, On the various decompositions of continuous and some weakly continuous, A cta
Math. Hungar., 71(1-2)(1996), 109-120.
[10] A. A. Atik, A study of some types of mapings on topological spaces, M.Sc. Thesis, Tanta Univ., Tanta, Egypt
(1997).
[11] A.I.EL-Maghrabi and A.A. Nasef, γ-open sets and its applications, Proc. Math. Phys. Soc. Egypt, 79(2003), 17-31.
[12] T. Fukutake; A. A. Nasef and A.I. EL-Maghrabi, Some topological concepts via γ- generalized closed sets,
Bulletin of Fukuoka University of Education, 52(III) (2003), 1-9.
[13] D.S. Jankovic and I. L. Reilly, On semi-separation properties, Indian J. Pure Appl. Math., 16(1985), 957-964.
[14] A. M. Kozae and A.A.EL-Ahmedy, On relations and topology, Preprint.
[15] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70(1963), 36-41.
[16] N. Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo, 2(19) (1970), 89-96.
[17] H. Maki; J. Umehara and T. Noiri, Every topological space is pre- T1/2 Mem. Fac. Sci. Koch. Univ. (Math.), 17
(1996), 33-42.
[18] A. S. Mashhour; M.E. Abd EL-Monsef and S.N.EL-Deeb, On precontinuous and weak precontinuous mappings,
Proc. Math. Phys. Soc. Egypt, 53(1982), 47-53.
[19] Z. Pawlak, Rough sets, Int. J. Inform. Comp. Sci., 11(1982), 341-356.
[20] Z. Pawlak, Rough sets, Rough relations and rough functions, Fundamental Informaticae, 27(1996), 103-108.
[21] H. Rasiwa and R. Sikorski, The Mathematics of Metamathematics, PWN, Warsaw (1968).
[22] A. Wiweger, On topological rough sets, Bull. Pol. AC. Math., 37(16) (1989), 89-93.
Source of support: Nil, Conflict of interest: None Declared
© 2013, IJMA. All Rights Reserved
187