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Mathematica
Balkanica
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New Series Vol. 23, 2009, Fasc. 1-2
Weakly b-Open Functions
1
Takashi Noiri 2 , Ahmad Al-Omari
Mohd. Salmi Md. Noorani 4
3
and
Presented by V. Kiryakova
We introduce a new class of functions called weakly b-open functions which is a generalization of weakly semi-open functions. Some characterizations and several properties concerning weakly b-open functions are obtained.
Key Words: b-open set, open function, b-open function , weakly b-open function
AMS Subj.Classification: 54C05, 54C08, 54C10
1. Introduction
In 1996, Andrijević [3] introduced a new class of generalized open sets
called b-open sets in a topological space. This class is a subset of the class of βopen sets [1]. Also the class of b-open sets is a superset of the class of semi-open
sets [8]. In [13], Rose defined the notion of weakly open functions in topological
spaces. Also the authors of [6,1] introduced and investigated weak forms of open
functions called semi-open functions and β-open functions. The purpose of this
paper is to present the class of weakly b-open functions as a new generalization
of weakly open functions. We investigate some of the fundamental properties
of this class of functions. For the benefit of the reader, we recall some basic
definitions and known results. Throughout the paper, the spaces X and Y (
or (X, τ ) and (Y, σ) ) stand for topological spaces with no separation axioms
assumed unless otherwise stated. Let A be a subset of X. The closure of A and
the interior of A will be denoted by Cl(A) and Int(A), respectively.
1
This work is financially supported by the Ministry of Higher Education, Malaysia under
FRGS grant no: UKM-ST-06-FRGS0008-2008.
2
T. Noiri, A.Al-Omari and Mohd. Salmi Md. Noorani
Definition 1.1. A subset A of a space X is said to be
1. α-open [9] if A ⊆ Int(Cl(Int(A)));
2. semi-open [8] if A ⊆ Cl(Int(A));
3. preopen [12] if A ⊆ Int(Cl(A));
4. β-open [1] or semi-preopen [2] if A ⊆ Cl(Int(Cl(A)));
5. b-open [3] if A ⊆ Cl(Int(A)) ∪ Int(Cl(A)).
The complement of an α-open ( resp. semi-open, preopen, b-open, βopen) set is called α-closed ( resp. semi-closed, preclosed, b-closed, β-closed ).
The smallest α-closed ( resp. semi-closed, preclosed, b-closed, β-closed) set containing A ⊆ X is called the α-closure ( resp. semi-closure, preclosure, b-closure,
β-closure ) of A and shall be denoted by αCl(A) (resp. sCl(A), pCl(A), bCl(A),
βCl(A)). The union of all α-open ( resp. semi-open, preopen, b-open, β-open)
sets of X contained in A is called the α-interior ( resp. semi-interior, preinterior, b-interior, β-interior ) of A and is denoted by αInt(A) (resp. sInt(A),
pInt(A), bInt(A), βInt(A)). The family of all b-open (resp. α-open, semi-open,
preopen, β-open, b-closed ) subsets of a space X is denoted by BO(X) (resp.
αO(X), SO(X), P O(X), βO(X), BC(X) ) and the collection of all b-open subsets of X containing a fixed point x is denoted by BO(X, x). The sets SO(X, x),
αO(X, x), P O(X, x), βO(X, x) and BC(X, x) are defined analogously. A point
x ∈ X is called a θ-cluster point of A if Cl(U ) ∩ A 6= φ for every open set U of X
containing x. The set of all θ-cluster points of A is called the θ-closure [15] of A
and is denoted by Clθ (A). A subset A is said to be θ-closed [15] if Clθ (A) = A.
The complement of a θ-closed set is said to be θ-open.
The following basic properties of the b-closure are useful in the sequel:
Lemma 1.2. [3] For a subset A of a space X, the following properties
hold:
1. bInt(A) = sInt(A) ∪ pInt(A);
2. bCl(A) = sCl(A) ∩ pCl(A);
3. bCl(X − A) = X − bInt(A);
4. x ∈ bCl(A) if and only if A ∩ U 6= φ for every U ∈ BO(X, x);
Weakly b-Open Functions
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5. A ∈ BC(X) if and only if A = bCl(A);
6. pInt(bCl(A)) = bCl(pInt(A));
Proposition 1.3.
properties are equivalent:
[3] For a subset A of a space X, the following
1. A is b-open;
2. A = pInt(A) ∪ sInt(A);
3. A ⊆ pCl(pInt(A)).
Lemma 1.4. [2] For a subset A of a space X, the following properties
hold:
1. αInt(A) = A ∩ Int(Cl(Int(A)));
2. sInt(A) = A ∩ Cl(Int(A));
3. pInt(A) = A ∩ Int(Cl(A)).
2. Weakly b-open functions
Definition 2.1.
A function f : X → Y is said to be b-open (resp.
α-open [11], semi-open [6], β-open [1] ) if the image of each open set U of X is
a b-open (resp. α-open, semi-open, β-open ) set.
Clearly, every open function is b-open, but the converse is not true.
E x a m p l e 2.2. Let X = {a, b, c, d}, τ = {X, φ, {b}, {a, b}, {b, c, d}},
and f : X → X be a function defined by f (a) = c, f (b) = b, f (c) = a and
f (d) = d. Then f is a b-open function, but it is not an open function since
f ({a, b}) = {b, c} which is b-open not open.
Theorem 2.3. A function f : X → Y is b-open if and only if for each
x ∈ X and each open set U of X containing x, there exists a b-open W ⊆ Y
containing f (x) such that W ⊆ f (U ).
P r o o f. The proof follows immediately from Definition 2.1.
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T. Noiri, A.Al-Omari and Mohd. Salmi Md. Noorani
For generalizations of open functions, the following diagram holds and
none of its implications is reversible:
open function ⇒ α-open function ⇒ semi-open function ⇒ b-open function ⇒
β-open function.
E x a m p l e 2.4. [3] Consider the set R of real numbers with the usual
topology, and let A = [0, 1] ∪ ((1, 2) ∩ Q), where Q stands for the set of rational
numbers. Then A is b-open but neither semi-open nor preopen. On the other
hand, let B = [0, 1) ∩ Q. Then B is β-open but not b-open.
E x a m p l e 2.5. Let R be the set of real numbers ,u is usual topology ,
τ = {φ, R, A} where A as Example 2.4, and f : (R, τ ) → (R, u) be the identity
function. Then f is b-open , but it is not semi-open.
E x a m p l e 2.6. Let R be the set of real numbers ,u is usual topology ,
τ = {φ, R, B} where B as Example 2.4, and f : (R, τ ) → (R, u) be the identity
function. Then f is β-open , but it is not b-open.
Theorem 2.7. If f : X → Y is b-open, then for any subset W ⊆ Y
and any closed set F ⊆ X containing f −1 (W ), there exists a b-closed set H ⊆ Y
containing W such that f −1 (H) ⊆ F .
P r o o f. Let H = Y −f (X −F ). Since f −1 (W ) ⊆ F , we have f (X −F ) ⊆
Y −W . Since f is b-open, then H is b-closed and f −1 (H) = X −f −1 (f (X −F )) ⊆
X − (X − F ) = F .
Corollary 2.8.
properties hold:
If f : X → Y is a b-open function, then the following
1. f −1 (Int(Cl(B)) ∩ Cl(Int(B))) ⊆ Cl(f −1 (B)) for each set B ⊆ Y .
2. f −1 (sInt(B)) ⊆ Cl(f −1 (B)) for each B ∈ P O(Y ).
3. f −1 (pInt(B)) ⊆ Cl(f −1 (B)) for each B ∈ SO(Y ).
P r o o f. (1) Since Cl(f −1 (B)) is closed in X containing f −1 (B) for a
set B ⊆ Y , then by Theorem 2.7, there exists a b-closed set H ⊆ Y , B ⊆ H
such that f −1 (H) ⊆ Cl(f −1 (B)). Since H is b-closed, thus f −1 (Int(Cl(B)) ∩
Cl(Int(B))) ⊆ f −1 (Int(Cl(H)) ∩ Cl(Int(H))) ⊆ f − (H) ⊆ Cl(f −1 (B)).
(2) and (3) they follow directly from (1) and Lemma 1.4.
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The relationship between the notions of b-open functions and open functions is given by the following result whose proof is an immediate consequence
of the definitions.
Thorem 2.9.
Suppose BO(Y, σ) is a topology on Y . A function
f : (X, τ ) → (Y, σ) is b-open if and only if f : (X, τ ) → (Y, BO(Y, σ)) is open.
We define the concept of weak b-openness.
Definition 2.10. A function f : X → Y is said to be weakly b-open if
f (U ) ⊆ bInt(f (Cl(U )))) for each open set U of X.
Clearly, every weakly open function is weakly b-open and every b-open
function is also weakly b-open, but the converse is not generally true.
E x a m p l e 2.11. Let D be the discrete topology on the positive real
numbers R+ and τ the usual topology on the real numbers R.
Define f : (R+ , D) → (R, τ ) by
x,
if x is rational ;
f (x) =
−x, if x is irrational .
Since f (U ) = f (Cl(U )) for any subset U of R+ , f is weakly b-open.
However f is not weakly open because Int(f (Cl(U ))) = φ for any subset U of
R+ .
E x a m p l e 2.12. Let X = {a, b, c} and τ = {X, φ, {a}, {b}, {a, b}, {a, c}}.
Then BO(X) = (X, τ ). Let f : X → X be a function defined by f (a) = c,
f (b) = b and f (c) = a. Then f is a weakly b-open function which is not b-open,
since for U = {a}, f (U ) is not b-open in X.
Theorem 2.13. If f : X → Y is weakly b-open and strongly continuous,
then f is b-open.
P r o o f. Let U be an open subset of X. Since f is weakly b-open, f (U ) ⊆
bInt(f (Cl(U ))). However, because f is strongly continuous, f (U ) ⊆ bInt(f (U ))
and therefore f (U ) is b-open.
E x a m p l e 2.14. A b-open function need not be strongly continuous.
Let X = {a, b, c} and τ = {X, φ, {a}, {b}, {a, b}, {a, c}}. Then the identity function f : X → X is b-open but it is not strongly continuous, since if U = {a},
f (Cl(U )) * f (U )
Recall that f : X → Y is said to be relatively weakly open [4] provided
that f (U ) is open in f (Cl(U )) for every open subset U of X.
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T. Noiri, A.Al-Omari and Mohd. Salmi Md. Noorani
Theorem 2.15.
open, then f is b-open.
If f : X → Y is weakly b-open and relatively weakly
P r o o f. Let U be an open subset of X and let y ∈ f (U ). Since f is relatively weakly open, there is an open subset V of Y for which f (U ) = f (Cl(U )) ∩
V . Because f weakly b-open, it follows that f (U ) ⊆ bInt(f (Cl(U ))). Then y ∈
bInt(f (Cl(U ))) ∩ V ⊆ f (Cl(U )) ∩ V = f (U ), and hence bInt(f (Cl(U ))) ∩ V =
f (U ). Since the intersection of an open set and a b-open set is b-open, therefore
f (U ) is b-open.
Now, we define an additional near b-open condition which implies bopenness if it is combined with weak b-openness.
Definition 2.16. A function f : X → Y is said to satisfy the weakly
b-open interiority condition if bInt(f (Cl(U ))) ⊆ f (U ) for every open subset U
of X.
Theorem 2.17 If a function f : X → Y is a weakly b-open and satisfies
the weakly b-open interiority condition, then f is b-open.
P r o o f. Let U be an open subset of X. Since f is weakly b-open, we have
that f (U ) ⊆ bInt(f (Cl(U ))). And by the weakly b-open interiority condition,
we have bInt(f (Cl(U ))) ⊆ f (U ). Hence bInt(f (Cl(U ))) = f (U ) and therefore
f (U ) is b-open in Y .
3. Characterizations of weakly b-open functions
Theorem 3.1. For a function f : X → Y , the following conditions are
equivalent:
1. f is weakly b-open;
2. f (Intθ (A)) ⊆ bInt(f (A)) for every subset A of X;
3. Intθ (f −1 (B)) ⊆ f −1 (bInt(B)) for every subset B of Y ;
4. f −1 (bCl(B)) ⊆ Clθ (f −1 (B)) for every subset B of Y .
P r o o f. (1) ⇒ (2): Let A be any subset of X and x ∈ Intθ (A). Then,
there exists an open set U such that x ∈ U ⊆ Cl(U ) ⊆ A. Then, f (x) ∈
f (U ) ⊆ f (Cl(U )) ⊆ f (A). Since f is weakly b-open, f (U ) ⊆ bInt(f (Cl(U ))) ⊆
bInt(f (A)).
This implies that f (x) ∈ bInt(f (A)). This shows that
x ∈ f −1 (bInt(f (A))). Thus, Intθ (A) ⊆ f −1 (bInt(f (A))), and so, f (Intθ (A)) ⊆
bInt(f (A)).
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(2) ⇒ (3): Let B be any subset of Y . Then by (2), f (Intθ (f −1 (B))) ⊆
bInt(f (f −1 (B))) ⊆ bInt(B). Therefore, Intθ (f −1 (B)) ⊆ f −1 (bInt(B)).
(3) ⇒ (4): Let B be any subset of Y . Using (3), we have
X − Clθ (f −1 (B)) = Intθ (X − f −1 (B))
= Intθ (f −1 (Y − B))
⊆ f −1 (bInt(Y − B))
= f −1 (Y − bCl(B))
= X − f −1 (bCl(B)).
Therefore, we obtain f −1 (bCl(B)) ⊆ Clθ (f −1 (B)).
(4) ⇒ (1): Let V be any open set of X and B = Y − f (Cl(V )). By (4),
f −1 (bCl(Y − f (Cl(V )))) ⊆ Clθ (f −1 (Y − f (Cl(V )))). Therefore, we obtain
f −1 (Y − bInt(f (Cl(V )))) ⊆ Clθ (X − f −1 (f (Cl(V )))) ⊆ Clθ (X − Cl(V )). Hence
V ⊆ Intθ (Cl(V )) ⊆ f −1 (bInt(f (Cl(V )))) and f (V ) ⊆ bInt(f (Cl(V ))). This
shows that f is weakly b-open.
Theorem 3.2. Let X be a regular space. Then for a function f : X →
Y , the following conditions are equivalent:
1. f is weakly b-open;
2. f is b-open;
3. For any set B of Y and any closed set A in X containing f −1 (B), there
exists a b-closed set F in Y containing B such that f −1 (F ) ⊆ A.
P r o o f. (1) ⇒ (2): Let V be a nonempty open subset of X. For each
x ∈ V , let Ux be an open set such that x ∈ Ux ⊆ Cl(Ux ) ⊆ V . Hence we obtain
that V = ∪{Ux : x ∈ V } = ∪{Cl(Ux ) : x ∈ V } and, f (V ) = ∪{f (Ux ) : x ∈
V } ⊆ ∪{bInt(f (Cl(Ux ))) : x ∈ V } ⊆ bInt(f (∪{Cl(Ux ) : x ∈ V }) = bInt(f (V )).
Thus f is b-open.
(2) ⇒ (3): This is same as Proof of Theorem 2.7.
(3) ⇒ (1): Let B be any set in Y . Let A = Cl(f −1 (B)). Then A is a closed
set in X and f −1 (B) ⊆ A. By (3) there exists a b-closed set F in Y containing
B such that f −1 (F ) ⊆ A. Since F is b-closed, f −1 (bCl(B)) ⊆ f −1 (F ) ⊆ A =
Cl(f −1 (B)) ⊆ Clθ (f −1 (B)). Therefore by Theorem 3.1, f is weakly b-open.
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Theorem 3.3. For a function f : X → Y , the following conditions are
equivalent:
1. f is weakly b-open;
2. For each x ∈ X and each open subset U of X containing x, there exists a
b-open set V containing f (x) such that V ⊆ f (Cl(U )).
P r o o f. (1) ⇒ (2): Let x ∈ X and U be an open set in X with
x ∈ U . Since f is weakly b-open, f (x) ∈ f (U ) ⊆ bInt(f (Cl(U ))). Let
V = bInt(f (Cl(U ))). Then V is b-open and f (x) ∈ V ⊆ f (Cl(U )).
(2) ⇒ (1): Let U be an open set in X and let y ∈ f (U ). It follows from (2)
that V ⊆ f (Cl(U )) for some b-open set V in Y containing y. Hence, we have
y ∈ V ⊆ bInt(f (Cl(U ))). This shows that f (U ) ⊆ bInt(f (Cl(U ))). Thus f is
weakly b-open.
Theorem 3.4.
For a bijective function f : X → Y , the following
conditions are equivalent:
1. f is weakly b-open;
2. bCl(f (Int(F ))) ⊆ f (F ) for each closed set F in X;
3. bCl(f (U )) ⊆ f (Cl(U )) for each open set U in X.
P r o o f. (1) ⇒ (2): Let F be a closed set in X. Then we have f (X −F ) =
Y − f (F ) ⊆ bInt(f (Cl(X − F ))), and so Y − f (F ) ⊆ Y − bCl(f (Int(F ))). Hence
bCl(f (Int(F ))) ⊆ f (F ) .
(2) ⇒ (3): Let U be an open set in X. Since Cl(U ) is a closed set and
U ⊆ Int(Cl(U )), by (2) we have bCl(f (U )) ⊆ bCl(f (Int(Cl(U )))) ⊆ f (Cl(U )).
(3) ⇒ (1): Let V b an open set of X. Then, we have Y − bInt(f (Cl(V ))) =
bCl(Y − f (Cl(V ))) = bCl(f (X − Cl(V ))) ⊆ f (Cl(X − Cl(V ))) = f (X −
Int(Cl(V ))) ⊆ f (X−V ) = Y −f (V ). Therefore, we have f (V ) ⊆ bInt(f (Cl(V )))
and hence f is weakly b-open..
The proof of the following theorem is straightforward and thus is omitted.
Theorem 3.5. For a function f : X → Y , the following conditions are
equivalent:
1. f is weakly b-open;
2. f (U ) ⊆ bInt(f (Cl(U ))) for each preopen set U of X;
Weakly b-Open Functions
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3. f (U ) ⊆ bInt(f (Cl(U ))) for each α-open set U of X;
4. f (Int(Cl(U ))) ⊆ bInt(f (Cl(U ))) for each open set U of X;
5. f (Int(F )) ⊆ bInt(f (F )) for each closed set F of X.
Lemma 3.6. [3] Let A be a subset of X. Then the following hold:
1. bCl(A) = sCl(A) ∩ pCl(A) = A ∪ [Cl(Int(A)) ∩ Int(Cl(A))],
2. bInt(A) = sInt(A) ∪ pInt(A) = A ∩ [Cl(Int(A)) ∪ Int(Cl(A))].
Theorem 3.7. For a function f : X → Y , the following conditions are
equivalent:
1. f is weakly b-open;
2. f (Intθ (A)) ⊆ [Cl(Int(f (A))) ∪ Int(Cl(f (A)))] for every subset A of X;
3. Intθ (f −1 (B)) ⊆ f −1 ([Cl(Int(B)) ∪ Int(Cl(B))]) for every subset B of Y ;
4. f −1 ([Cl(Int(B)) ∩ Int(Cl(B))]) ⊆ Clθ (f −1 (B)) for every subset B of Y ;
5. f (Int(F )) ⊆ [Cl(Int(f (F ))) ∪ Int(Cl(f (F )))] for each closed set F of X;
6. f (Int(Cl(U ))) ⊆ [Cl(Int(f (Cl(U )))) ∪ Int(Cl(f (Cl(U ))))] for each open
set U of X;
7. f (U )) ⊆ [Cl(Int(f (Cl(U )))) ∪ Int(Cl(f (Cl(U ))))] for each preopen set U
of X;
8. f (U )) ⊆ [Cl(Int(f (Cl(U )))) ∪ Int(Cl(f (Cl(U ))))] for each α-open set U
of X.
P r o o f. This follows from Theorems 3.1,3.5 and 3.6
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T. Noiri, A.Al-Omari and Mohd. Salmi Md. Noorani
4. Some properties of weakly b-open functions
Theorem 4.1. If a function f : X → Y is weakly b-open and continuous
, then f is β-open.
P r o o f. Let U be an open subset of X. Then by weak b-openness of f ,
f (U ) ⊆ bInt(f (Cl(U ))). Since f is continuous, f (Cl(U )) ⊆ Cl(f (U )). Hence
we obtain that,
f (U ) ⊆ bInt(f (Cl(U )))
⊆ bInt(Cl(f (U )))
= sInt(Cl(f (U ))) ∪ pInt(Cl(f (U ))))
⊆ Cl(Int(Cl(f (U )))) ∪ Int(Cl(f (U )))
⊆ Cl(Int(Cl(f (U ))))
which shows that f (U ) is a β-open set in Y . Thus f is a β-open function.
Definition 4.2. [5] A function f : X → Y is said to be contra-open
(resp. contra-closed ) if f (U ) is closed ( resp. open ) in Y for each open ( resp.
closed ) subset U of X.
Since every semi-open set is b-open, then every semi-open function is
b-open.
Theorem 4.3.
then f is semi-open.
If f : X → Y is a b-open and contra-open function,
P r o o f. Let U be an open subset of X. Since f is b-open and contra-open,
f (U ) = bInt(f (U ))
= sInt(f (U )) ∪ pInt(f (U )))
⊆ Cl(Int(f (U )))) ∪ Int(Cl(f (U )))
⊆ Cl(Int(f (U ))) ∪ Int(f (U ))
= Cl(Int(f (U ))).
Hence f is a semi-open function.
Theorem 4.4. If f : X → Y is a weakly b-open bijective function of a
space X onto a b-connected space Y , then X is connected.
P r o o f. Suppose that X is not connected. Then there exist non-empty
open sets U and V such that U ∩ V = φ and U ∪ V = X. Hence we have
f (U ) ∩ f (V ) = φ and f (U ) ∪ f (V ) = Y . Since f is weakly b-open, we have
Weakly b-Open Functions
11
f (U ) ⊆ bInt(f (Cl(U ))) and f (V ) ⊆ bInt(f (Cl(V ))). Moreover U , V are open
and also closed. We have f (U ) = bInt(f (Cl(U ))) and f (V ) = bInt(f (Cl(V ))).
Hence, f (U ) and f (V ) are b-open in Y . Thus, Y has been decomposed into two
non-empty disjoint b-open sets. This is contrary to the hypothesis that Y is a
b-connected space. Thus, X is connected.
Recall that a subset K of a space X is said to be quasi H-closed [15] (
resp. b-compact [7] ) relative to X if each cover of K by open ( resp. b-open )
sets of X contains a finite subcollection whose closures cover the set ( resp. has
a finite subcollection which covers the set ).
Theorem 4.5. Let f : X → Y be bijective weakly b-open and K a set
of Y which is b-compact relative to Y , then f −1 (K) is quasi H-closed relative to
X.
P r o o f. Let Λ = {Ui : i ∈ I} be a cover of f −1 (K) by open sets of X,
where I is an index set. For each y ∈ K, f −1 (y) ∈ Ui(y) for some Ui(y) ∈ Λ.
By weak b-openness of f , there exists a b-open set Wy containing y such that
Wy ⊆ bInt(f (Cl(Uy ))). The collection {Wy : y ∈ K} a cover of K by b-open
sets of Y and so there is a finite subcover {Wy : y ∈ K0 }, where K0 is a finite
subset of K. Clearly {Cl(Uy ) : y ∈ K0 } covers f −1 (K), i.e. f −1 (K) is quasi
H-closed relative to X.
Definition 4.6. [10] A space X is said to be hyperconnected if every
non-empty open subset of X is dense in X.
Theorem 4.7.
Let X be a hyperconnected space. Then a function
f : X → Y is weakly b-open if and only if f (X)is b-open in Y .
P r o o f. The sufficiency is clear. For the necessity observe that for any
open subset U of X, f (U ) ⊆ f (X) = bInt(f (X)) = bInt(f (Cl(U ))). Hence f is
weakly b-open.
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2949-1 Shiokita-cho, Hinagu,
Yatsushiro-shi, Kumamoto-ken
869-5142 JAPAN
E-mail: [email protected]
Received 24.04.2008
Weakly b-Open Functions
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Department of Mathematics and Statistics,
Faculty of Science,
Mu’tah University
P.O.Box 7
Karak-JORDAN
E-mail: [email protected]
4
School of Mathematical Sciences
Faculty of Science and Technology
Universiti Kebangsaan Malaysia
43600 UKM Bangi
Selangor, MALAYSIA
E-mail: [email protected]
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