International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013

ISSN 2229-5518

ON ALMOST STRONGLY θ-b-CONTINUOUS FUNCTIONS

HAKEEM A. OTHMAN∗ AND ALI TAANI∗∗

Abstract. We introduce a new class of functions called almost strongly θ-b-continuous function which is a generalization of strongly θ-continuous functions and strongly θ-b- continuous functions. Some characterizations and several properties concerning almost strongly θ-b-continuous function are obtained.

1. introduction
A subset A of a topological space X is b- open [2] or sp-open [7] if A ⊆ I nt(C l(A)) ∪ C l(I nt(A)). A function f : X → Y is called b-continuous [8] if for each x ∈ X and each open set V of Y containing f (x), there exists a b-open U containing x such that f (U ) ⊆ V , which is equivalent to say that the preimage
The complement of an b-open set is called

b-closed. The smallest b-closed set contain- ing A ⊆ X is called the b-closure, of A and shall be denoted by bC l(A). The union of all b-open set of X contained in A is called the b-interior of A and is denoted by bI nt(A). A subset A is said to be b-regular if it is b- open and b-closed. The family of all b-open (

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f −1(V ) of each open set V of Y is b-open in

X . Recently, Park [16] introduced and inves- tigated the notion of strongly θ-b-continuous functions which is stronger than b-continuous, moreover see [3, 4, 5].The purpose of the present paper is to introduce and investi- gate a weaker form of strongly θ-b-continuity called almost strongly θ-b-continuous func- tion.

For the benefit of the reader we recall some basic definitions and known results. Throughout the present paper, the space X and Y (or (X, τ ) and (Y, σ) ) stand for topo- logical spaces with no separation axioms as- sumed, unless otherwise stated. Let A be a subset of X . The closure of A and the interior of A will be denoted by C l(A) and I nt(A), respectively.
resp; b-closed, b-regular, open ) subsets of a
space X is denoted by BO(X ) ( resp; BC (X ), BR(X ),O(X ) respectively ) and the collec- tion of all b-open subsets of X containing a fixed point x is denoted by BO(X, x). The sets O(X, x) and BR(X, x) are defined anal- ogously.
A point x ∈ X is called a θ-cluster point of

A if C l(U ) ∩ A = φ for every open set U of X containing x. The set of all θ-cluster points of A is called the θ-closure [18] of A and is denoted by C lθ (A). A subset A is said to be θ-closed [18] if C lθ (A) = A. The complement of a θ-closed set is said to be θ-open.

A point x of X is called a b-θ-cluster [16] point of A if bC l(U ) ∩ A = φ for every U ∈ BO(X, x). The set of all b-θ-cluster points of A is called b-θ-closure of A and is denoted by bC lθ (A). A subset A is said to be

2000 Mathematics Subject Classification. 54C05, 54C08, 54C10 .

Key words and phrases. b-open sets, b-θ-closed sets, Almost strongly θ-b-continuous functions, b-θ-closed

graphs.

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b-θ-closed if A = bC lθ (A). The complement of a b-θ-closed set is said to be b-θ-open set. A subset A of X is called regular open (regular closed) if A = I nt(C l(A)) (A = C l(I nt(A))). The δ-interior of a subset A of X is the union of all regular open sets of X contained in A and it is denoted by δ-I nt(A) [18]. A subset A is called δ-open if A = δ-I nt(A). The complement of a δ- open set is called δ-closed. The δ-closure of a set A in a space (X, τ ) is defined by

{x ∈ X : A ∩ I nt(C l(B)) = φ, B ∈ τ and

x ∈ B} and it is denoted by δ-C l(A).

2. characterizations Definition 2.1. A function f : X → Y is said to be almost strongly θ-b-continuous if
for each x ∈ X and each open set V of Y
containing f (x), there exists U ∈ BO(X, x)
such that f (bC l(U )) ⊆ I nt(C l(V )).

it is not strongly θ-b-continuous . Since the open set V = {a} in (X, σ) containing f (c)

and there is no b-open set U in (X, τ ) con- taining c such that f (bC l(U )) ⊆ V .

Theorem 2.5. For a function f : X → Y , the following are equivalent:

(1) f is almost strongly θ-b-continuous;
(2) f −1(V ) is b-θ-open in X for each reg- ular open set V of Y ;
(3) f −1(F ) is b-θ-closed in X for each reg- ular closed set F of Y ;
(4) for each x ∈ X and each regular open set V of Y containing f (x), there exists U ∈ BO(X, x) such that f (bC l(U )) ⊆ V ;
(5) f −1(V ) is b-θ-open in X for each δ- open set V of Y ;
(6) f −1(F ) is b-θ-closed in X for each δ- closed set F of Y ;
(7) f (bC lθ (A)) ⊆ C lδ (f (A)) for each sub-

set A of X ;

1(B)) ⊆ f
1(C lδ (B)) for each
Definition 2.2. [16] A function f : X → Y is said to be strongly θ-b-continuous if for each x ∈ X and each open set V of Y con- taining f (x), there exists U ∈ BO(X, x) such that f (bC l(U )) ⊆ V .
Then it is clear that every strongly θ-b- continuous is almost strongly θ-b-continuous but the converse is not true.
Definition 2.3. [14] A function f : X → Y
is said to be strongly θ-continuous if for each x ∈ X and each open set V of Y containing f (x), there exists an open set U of X con- taining x such that f (C l(U )) ⊆ V .

Example 2.4. Let X = {a, b, c},

(X, τ ) = {X, φ, {a}, {a, b}} with BO(X, τ ) =

{X, φ, {a}, {a, b}, {a, c}} and (X, σ) =

{X, φ, {a}}. And f : (X, τ ) → (X, σ) be defined by f (a) = b, f (b) = c and f (c) = a. Then f is almost strongly θ-b-continuous but

(8) bC lθ (f − −

subset B of Y .

Proof. (1) → (2): Let V be any regular open set of Y and x ∈ f −1(V ). Then V = int(clV ) and f (x) ∈ V . Since f is almost strongly θ-b- continuous, there exists U ∈ BO(X, x) such that f (bC l(U )) ⊆ V . Therefore, we have x ∈ U ⊆ bC l(U ) ⊆ f −1(V ). This shows that f −1(V ) is b-θ-open in X .

(2) → (3): Let F be any regular closed set of Y . By (2), f −1(F ) = X − f −1(Y − F ) is b-θ-closed in X .
(3) → (4): Let x ∈ X and V be any regu- lar open set of Y containing f (x). By (3), f −1(Y − V ) = X − f −1(V ) is b-θ-closed in X and so f −1(V ) is a b-θ-open set con- taining x, there exists U ∈ BO(X, x) such that bC l(U ) ⊆ f −1(V ). Therefore, we have f (bC l(U )) ⊆ V .
(4) → (5): Let V be any δ-open set of Y

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and x ∈ f −1(V ). There exists a regular open set G of Y such that f (x) ∈ G ⊆ V . By (4), there exists U ∈ BO(X, x) such that f (bC l(U )) ⊆ G. Therefore, we obtain x ∈ U ⊆ bC l(U ) ⊆ f −1(V ). This shows that f −1(V ) is b-θ-open in X .
(5) → (6): Let F be any δ-closed set of Y . Then Y − F is b-θ-open in Y and by (5), f −1(F ) = X − f −1(Y − F ) is b-θ-closed in X .
(6) → (7): Let A be any subset of X . Since C lδ (f (A)) is δ-closed in Y , by (6)

f −1(C lδ (f (A))) is b-θ-closed in X . Let x ∈/

(2) bC lθ (f −1(V ) ⊆ f −1(C l(V )) for each

β-open set V of Y ;

(3) bC lθ (f −1(V ) ⊆ f −1(C l(V )) for each

b-open set V of Y ;

(4) bC lθ (f −1(V ) ⊆ f −1(C l(V )) for each

semi-open set V of Y .

Proof. (1) → (2): Let V be any β-open set of Y . Then by Theorem 2.4 in [1] C l(V ) is reg- ular closed in Y . Since f is almost strongly θ-b-continuous, f −1(C l(V )) is b-θ-closed in X

and hence bC lθ (f −1(V ) ⊆ f −1(C l(V )).
(2) → (3): This is obvious since every b-open
set is β-open.

f −1(C lδ (f (A))). There exists U ∈ BO(X, x)

such that bC l(U ) ∩ f −1(C lδ (f (A))) = φ and
(3)
op

→ (4): This is obvious since every semi- is b-open.

thus bC l(U ) ∩ A = φ. Hence x ∈/

bC lθ (A).

en set
(4) → (1): Let F be any regular closed set
Therefore, we have f (bC lθ (A)) ⊆ C lδ (f (A)).
(7) → (8): Let B be any subset of Y . By
of Y . Then F is semi-open in Y and by(4)

bC lθ (f −1(F ) ⊆ f −1(C l(F )) = f −1(F ). This

(7), we have f (bC lθ (f −1(B))) ⊆ C lδ (B) and
shows that f
−1(F ) is b-θ-closed in X . There-
hence bC lθ (f −1(B)) ⊆ f −1(C lδ (B)).
fore f is almost strongly θ-b-continuous.

(8) → (1): let xI∈ JX and VSbe any ER D

open set of Y containing f (x). Then G = Y − I nt(C l(V )) is regular closed and hence δ-closed in Y . By (8), bC lθ (f −1(G)) ⊆ f −1(C lδ (G)) = f −1(G) and hence f −1(G) is b-θ-closed in X . Therefore, f −1(I nt(C l(V )))
is b-θ-open set containing x. There ex- ists U ∈ BO(X, x) such that bC l(U ) ⊆ f −1(I nt(C l(V ))). Therefore we obtain f (bC l(U )) ⊆ I nt(C l(V )). This shows that f is almost strongly θ-b-continuous. D
Definition 2.6. A subset A of a space X is said to be:
(1) α-open [12] if A ⊆ I nt(C l(I nt(A))); (2) semi-open [9] if A ⊆ C l(I nt(A));
(3) preopen [11] if A ⊆ I nt(C l(A)); (4) β-open [2] if A ⊆ C l(I nt(C l(A))).

Theorem 2.7. For a function f : X → Y , the following are equivalent:

(1) f is almost strongly θ-b-continuous;
Recall that a space X is said to be almost
regular [15](resp; semi-regular) if for any reg- ular open (resp; open ) set U of X and each point x ∈ U , there exist a regular open set V of X such that x ∈ V ⊆ C l(V ) ⊆ U (resp; x ∈ V ⊆ U ).

Theorem 2.8. For any function f : X → Y , the following properties hold:

(1) If f is b-continuous and Y is almost regular, then f is almost strongly θ-b- continuous;
(2) If f is almost strongly θ-b-continuous and Y is semi-regular, then f is strongly θ-b-continuous;

Proof. (1) Let x ∈ X and V be any regu- lar open set of Y containing f (x). Since Y is almost regular, there exists an open set W such that f (x) ∈ W ⊆ C l(W ) ⊆ V . Since f is b-continuous, there exists U ∈ BO(X, x) such that f (U ) ⊆ W . We shall

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show that f (bC l(U )) ⊆ C l(W ). Suppose that

X is b-regular then f is almost strongly θ-b-

y ∈/

C l(W ). There exists an open neighbor-

continuous.

hood G of y such that G ∩ W = φ. Since

f is b-continuous, f −1(G) ∈ BO(X ) and f −1(G)∩U = φ and hence f −1(G)∩bC l(U ) = φ. Therefore, we obtain G ∩ f (bC l(U )) = φ and y ∈/ f (bC l(U )). Consequently. we have f (bC l(U )) ⊆ C l(W ) ⊆ V .

(2) Let x ∈ X and V be any open set of Y containing f (x). Since Y is semi-regular, there exists a regular open set W such that f (x) ∈ W ⊆ V . Since f is almost strongly θ- b-continuous, there exists U ∈ BO(X, x)such that f (bC l(U ) ⊆ W . Therefore, we have f (bC l(U ) ⊆ V . D
Definition 2.9. A topological space X is

Proof. (1) Let f : X → Y be the iden- tity. Then f is continuous and hence almost strongly θ-b-continuous. For any regular open set U of X and any points x ∈ U , we have f (x) = x ∈ U and there exists G ∈ BO(X, x) such that f (bC l(G)) ⊆ U . Therefore, we have x ∈ G ⊆ bC l(G) ⊆ U and hence X is almost b-regular.

(2) Suppose that f : X → Y is almost continuous and X is b-regular. For each x ∈ X and any regular open set V contain- ing f (x), f −1(V ) is an open set of X con- taining x. Since X is b-regular there exists U ∈ BO(X, x) such that x ∈ U ⊆ bC l(U ) ⊆
1(V ). Therefore, we have f (bC l(U )) ⊆ V .
said to be b∗-regular ( resp; b-regular [16], al- most b-regular ) if for each F ∈ BC (X ) ( resp; F ∈ C (X ),F regular closed ) and each x ∈/ F , there exist disjoint b-open sets U and

f −

This shows that f is almost strongly θ-b- continuous. D Theorem 2.12. [16] Let A and B be any

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V such that x ∈ U and F ⊆ V .

Lemma 2.10. For a topological space X , the following are equivalent:

subset of a space X . Then the following prop-

erties hold:

(1) A ∈ BR(X ) if and only if A is b-θ- open and b-θ-closed;

(1) X is b∗-regular ( resp; b-regular [16] );

(2) For each U ∈ BO(X, x) ( resp; U ∈
(2) x

∈ bC lθ (A) if and only if V

∩ A = φ

O(X, x) ) , there exists V ∈ BO(X, x)

such that x ∈ V ⊆ bC l(V ) ⊆ U .

It is Known that a function f : X → Y is almost continuous if for each x ∈ X and each open set V of Y containing f (x), there is a neighborhood U of x such that f (U ) ⊆ I nt(C l(V )). Long and Herrington [10] proved that f : X → Y is almost continuous if and only if the inverse image of every regular open set in Y is open in X .

Theorem 2.11. (1) If a continuous function f : X → Y is almost strongly θ-b-continuous then X is almost b-regular.

(2) If f : X → Y is almost continuous and

for each V ∈ BR(X, x);

(3) A ∈ BO(X ) if and only if bC l(A) ∈

BR(X );

(4) A ∈ BC (X ) if and only if bI nt(A) ∈

BR(X );

(5) A ∈ BO(X ) if and only if bC l(A) =

bC lθ (A);

(6) A is b-θ-open in X if and only if for each x ∈ A there exists V ∈ BR(X ) such that x ∈ V ⊆ A.

Lemma 2.13. A subset U of a space X is b-θ-open in X if and only if for each x ∈ U , there exists b-open set W with x ∈ W such that bC l(W ) ⊆ U .

Theorem 2.14. For a function f : X → Y , the following are equivalent:

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(1) f is almost strongly θ-b-continuous;
(2) for each x ∈ X and each regular open set V of Y containing f (x), there ex- ists a b-θ-open set U containing x such that f (U ) ⊆ V ;
(3) for each x ∈ X and each regular open set V of Y containing f (x), there ex- ists a b-open set W containing x such

that f (bC lθ (W )) ⊆ V .

Proof. (1) → (2): Let x ∈ X and let V be any regular open subset of Y with f (x) ∈ V . Since f is almost strongly θ-b-continuous,

containing g(x). Since g is almost strongly θ- b-continuous there exists U ∈ BO(X, x) such that g(bC l(U )) ⊆ X × V . Therefore, we ob- tain f (bC l(U )) ⊆ V . Next we show that X is
almost b-regular. Let U be any regular open set of X and x ∈ U . Since g(x) ∈ U × Y and U × Y is regular open in X × Y , there exists G ∈ BO(X, x) such that g(bC l(G)) ⊆ U × Y . Therefore, we obtain x ∈ G ⊆ bC l(G) ⊆ U and hence X is almost b-regular.
(2) Let x ∈ X and W be any regular open set of X × Y containing g(x). there exist reg-

ular open sets U1 ⊆ X and V ⊆ Y such that

f −1(V ) is b-θ-open in X and x ∈ f −1(V ).

Let U = f −1(V ). Then f (U ) ⊆ V .

g(x) = (x, f (x))

∈ U1 × V

⊆ W . Since f is

(2) → (3): Let x ∈ X and let V be any reg- ular open subset of Y with f (x) ∈ V . By (2), there exists a b-θ-open set U containing x such that f (U ) ⊆ V . From Lemma 2.13 there exists a b-open set W such that x ∈ W ⊆

almost strongly θ-b-continuous, there exists U2 ∈ BO(X, x) such that f (bC l(U2)) ⊆ V . Since X is b∗-regular and U1 ∩U2 ∈ BO(X, x), there exists U ∈ BO(X, x) such that x ∈ U ⊆ bC l(U ) ⊆ U1 ∩ U2 (by Lemma 2.10). Therefore, we obtain g(bC l(U )) ⊆ U1 ×

IJSER

bC l(W ) ⊆ U . Since W is b-open, bC l(W ) =

bC lθ (W ), and then we have f (bC lθ (W )) ⊆ V . (3) → (1): This follows from Lemma 2.12(5).

D
3. some properties

Theorem 3.1. Let f : X → Y be a function and g : X → X × Y be the graph function of

f (bC l(U2)) ⊆ U1 × V ⊆ W . This shows that

g is almost strongly θ-b-continuous. D

Lemma 3.2. [13] If X0 is α-open in X , then

BO(X0) = BO(X ) ∩ X0.

Lemma 3.3. [16] If A ⊆ X0 ⊆ X , and X0 is

α-open in X , then bC l(A) ∩ X0 = bC lX0 (A),

f . Then, the following properties hold:

where bC l

(A) denotes the b-closure of A in
(1) If g is almost strongly θ-b-continuous, then f is almost strongly θ-b- continuous and X is almost b-regular;

(2) If f is almost strongly θ-b-continuous and X is b∗-regular, then g is almost

strongly θ-b-continuous.

Proof. (1) Suppose that g is almost strongly θ-b-continuous. First we show that f is al- most strongly θ-b-continuous. Let x ∈ X and V be a regular open set of Y containing f (x).

the subspace X0.

Theorem 3.4. If f : X → Y is almost strongly θ-b-continuous and X0 is a α-open

subset of X , then the restriction f |X0 : X0 →

Y is almost strongly θ-b-continuous.

Proof. For any x ∈ X0 and any regular open set V of Y containing f (x), there ex-

ists U ∈ BO(X, x) such that f (bC l(U )) ⊆ V
since f is almost strongly θ-b-continuous. Put

U0 = U ∩ X0, then by Lemmas 3.2 and 3.3,

Then X × V is a regular open set of X × Y

U0 ∈ (BO X)0,axnd b0C( lX ) U0

⊆ (bC )l .U0

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Therefore, we obtain (f |X0)(bC lX0 (U0)) =

f (bC lX0 (U0)) ⊆ f (bC l(U0)) ⊆ f (bC l(U )) ⊆

V . This shows that f |X0 is almost strongly

θ-b-continuous. D

Definition 3.5. A space X is said to be b- T2 ( resp; b-Urysohn ) [6] if for each pair of distinct points x and y in X , there exist U ∈ BO(X, x) and V ∈ BO(X, x) such that U ∩ V = φ (resp; bC l(U ) ∩ bC l(V ) = φ ).
Definition 3.6. A space X is said to be rT0 [1] if for each pair of distinct points x and y in X , there exist regular open set con- taining one of the points but not the other.

Theorem 3.7. Let f : X → Y be injective

D Lemma 3.8. Let A be a subset of X and B be a subset of Y . Then
(1) [13] If A ∈ BO(X )and B ∈ BO(Y ),

then A × B ∈ BO(X × Y ).

(2) [16] bC l(A × B) ⊂ bC l(A) × bC l(B).

Theorem 3.9. Let f : X1 → Y , g : X2 → Y

be two almost strongly θ-b-continuous and Y is Hausdorff, then A = {(x1, x2) : f (x1) = g(x2)} is b-θ-closed in X1 × X2.

Proof. Let (x1, x2) ∈/ A. Then f (x1) = g(x2). Since Y is Hausdorff, there exist open sets V1 and V2 containing f (x1) and g(x2) re-

spectively, such that V1 ∩ V2 = φ, hence

I nt(C l(V )) ∩ I nt(C l(V )) = φ. Since f and g

and almost strongly θ-b-continuous. 1 2

(1) If Y is rT0 , then X is b-T2;

(2) If Y is Hausdorff, then X is b- Urysohn.

are almost strongly θ-b-continuous, there ex- ists U1 ∈ BO(X, x1) and U2 ∈ BO(X, x2) such that f (bC l(U1)) ⊆ I nt(C l(V1)) and g(bC l(U2)) ⊆ I nt(C l(V2)). Since (x1, x2) ∈

IJSER

Proof. (1) Let x and y be any distinct points

of X . Since f is injective, f (x) = f (y) and there exists a regular open set V con- taining f (x) not containing f (y) or a regular open set W containing f (y) not containing f (x). If the first case holds, then there ex- ists U ∈ BO(X, x) such that f (bC l(U )) ⊆ V .

U1 × U2 ∈ BO(X1 × X2) and bC l(U1 × U2) ∩

A ⊆ (bC l(U1) × bC (U2)) ∩ A = φ, we have

that (x1, x2) ∈/ bC lθ (A). Thus A is b-θ-closed

in X1 × X2. D

In [13], Nasef introduced the notion of B∗- space. If for each x ∈ X , BO(X, x) is closed under finite intersection, then the space X is
Therefore, we obtain f (y) ∈/

f (bC l(U )) and

called B∗-space.
hence X − bC l(U ) ∈ BO(X, y). If the second
case holds, then we obtain a similar result.

Theorem 3.10. Let f , g be two almost

Therefore, X is b-T2.

strongly θ-b-continuous from a B

∗-space X

(2) As in (1), if x and y are distinct points of X , then f (x) = f (y). Since Y is Hausdorff, there exists open sets V and W containing

into a Hausdorff, space Y . Then the set

A = {x ∈ X : f (x) = g(x)} is b-θ-closed.

Proof. We will show that X \A is b-θ-open.

f (x) and f (y) respectively, such that V ∩W =

Let x ∈/

A, then f (x) = g(x). Since Y is

φ. Hence I nt(C l(V )) ∩ I nt(C l(W )) = φ. Since f is almost strongly θ-b-continuous, there exist G ∈ BO(X, x) and H ∈ BO(X, y) such that f (bC l(G)) ⊆ I nt(C l(V )) and f (bC l(H )) ⊆ I nt(C l(W )). It follows that bC l(G) ∩ bC l(H ) = φ. This shows that X is b-Urysohn.

Hausdorff, there exist open sets V1 and V2 in Y such that f (x) ∈ V1 and g(x) ∈ V2 and V1 ∩ V2 = φ, hence I nt(C l(V1)) ∩ I nt(C l(V2)) =

φ. Since f and g are almost strongly θ-b- continuous, there exist b-open sets U1 and

U2 containing x such that f (bC l(U1)) ⊆

I nt(C l(V1)) and g(bC l(U2)) ⊆ I nt(C l(V2)).

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Take U = U1 ∩ U2. Clearly U ∈ BO(X, x) be-

cause X is B∗-space and x ∈ U ⊆ bC l(U ) ⊆

Proof. Suppose that A is not b-θ-closed. Then there exists a point x in X such that

bC l(U1 ∩ U2) ⊆ bC l(U1) ∩ bC l(U2) ⊆ X \A

x ∈ bC lθ (A) but x ∈/

A. It follows that

because f (bC l(U1)) ∩ g(bC l(U2)) = φ. Thus

X \A is b-θ-open. D

Recall that for a function f : X → Y , the subset {(x, f (x)) : x ∈ X } of X × Y is called the graph of f and is denoted by G(f ).
Definition 3.11. The graph G(f ) of a func- tion f : X → Y is said to be b-θ-closed if for each (x, y) ∈ (X × Y ) \ G(f ), there exist U ∈ BO(X, x) and an open set V in Y containing y such that (bC l(U ) × C l(V )) ∩ G(f ) = φ.

r(x) = x because r is retraction. Since X is Hausdorff, there exist open sets U and V containing x and r(x) respectively, such that

U ∩ V = φ, hence sC l(U ) ∩ I nt(C l(V )) ⊆

C l(U ) ∩ I nt(C l(V )) = φ. By hypothe-

sis, there exists U∗ ∈ BO(X, x) such that r(bC l(U∗)) ⊆ I nt(C l(V )). Since U ∩ U∗ ∈ BO(X, x) and x ∈ bC lθ (A), we have we have bC l(U ∩ U∗) ∩ A = φ. Therefore, there exists a point y ∈ bC l(U ∩ U∗) ∩ A. So y ∈ A and r(y) = y ∈ bC l(U ). Since bC l(U ) = sC l(U ),

Lemma 3.12. The graph G(f ) of a func- tion f : X → Y is b-θ-closed if and only if for each (x, y) ∈ (X × Y ) \ G(f ), there exist U ∈ BO(X, x) and an open set V in Y con- taining y such that f (bC l(U )) ∩ C l(V ) = φ.

Theorem 3.13. Let f : X → Y be almost

sC l(U ) ∩ I nt(C l(V )) = φ gives r(y) ∈/ I nt(C l(V )). On the other hand, y ∈ bC l(U∗)

and this implies r(bC l(U∗)) � I nt(C l(V )).
This is contradiction with the hypothesis that

r is almost strongly θ-b-continuous retraction. Thus A is b-θ-closed subset of X . D

strongly θ-b-continIuous Jand Y is SHausdorff, ER

Theorem 3.15. Let X ,X1 and X2 be topo-

then G(f ) is b-θ-closed in X × Y .

logical spaces, If h : X → X1

× X2, h(x) =

Proof. Let (x, y) ∈ (X × Y ) \ G(f ). Then

f (x) = y Since Y is Hausdorff, there exists open sets V and W in Y containing f (x) and y respectively, such that I nt(C l(V )) ∩ C l(W ) = φ. Since f is almost strongly θ-b-continuous, there exist U ∈ BO(X, x) such that f (bC l(G)) ⊆ I nt(C l(V )). There- fore, f (bC l(U )) ∩ C l(W ) = φ. and then by

(x1, x2) is almost strongly θ-b-continuous then

fi : X → Xi, fi(x) = xi is almost strongly θ-

b-continuous for i = 1, 2.

Proof. We show only that f1 : X → X1 is al- most strongly θ-b-continuous. Let V1 be any

regular open set in X1. Then V1 × X2 is regu-

lar open in X1 ×X2 and hence h−1(V1 ×X2) is

Lemma 3.12 G(f ) is b-θ-closed in X × Y .

b-θ-open in X . Since f −1 1 1 2

D
Recall that a subspace A of X is called a re- tract of if there is a continuous map r : X →
1 (V ) = h−1(V

f1 is almost strongly θ-b-continuous.

×X ),

A (called a retraction) such that for all x ∈ X

and all a ∈ A, r(x) ∈ A, and r(a) = a.

Theorem 3.14. Let A be a subset of X and r : X → A be almost strongly θ-b-continuous retraction. If X is Hausdorff, then A is b-θ- closed subset of X .

A subset K of a space X is said to be b-
closed relative to X [16] ( resp; N -closed rela- tive to X [15]) if for every cover {Vα : α ∈ Λ}
of K by b-open ( regular open ) sets of X , there exists a finite subset Λ0 of Λ such that

K ⊆ ∪{bC l(Vα) : α ∈ Λ0} ( resp; K ⊆ ∪{Vα :

α ∈ Λ0}).

International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013

ISSN 2229-5518

399

Theorem 3.16. If a function f : X → Y

is almost strongly θ-b-continuous and K is b-closed relative to X ,then f (K ) is N -closed relative to Y .

Proof. Let {Vα : α ∈ Λ} be a cover of f (K )

by regular open sets of Y . For each point x ∈ K , there exists α(x) ∈ Λ such that f (x) ∈ Vα(x). Since f is almost strongly θ-b-continuous there exists Ux ∈ BO(X, x) such that f (bC l(Ux)) ⊆ Vα(x). The family

{Ux : x ∈ K } is a cover of K by b-open sets of

X and hence there exists a finite subset K0 of K such that K ⊆ ∪x∈K0 bC l(Ux). Therefore, we obtain f (K ) ⊆ ∪x∈K0 Vα(x). This shows

that f (K ) is N -closed relative to Y . D
A topological space X is said to be quasi-H - closed [17] if every cover of X by open sets has a finite subcover whose closures cover X .

Theorem 3.18. If a function f : X → Y has

a b-θ-closed graph, then f (K ) is θ-closed in Y for each subset K which is b-closed relative to X .

Proof. Let K be a b-closed relative to X and y ∈ Y \ f (K ). Then for each x ∈ K we have (x, y) ∈/ G(f ) and by Lemma 3.12, there exist Ux ∈ BO(X, x) and open set Vx of Y contain- ing y such that f (bC l(Ux)) ∩C l(Vx) = φ. The family {Ux : x ∈ K } is a cover of K by b-open

sets of X . Since K is b-closed relative to X , there exists a finite subset K0 of K such that

K ⊆ ∪{bC l(Ux : x ∈ K0)}. put V = ∩{Vx :

x ∈ K0}. then V is an open set containing

y and f (K ) ∩ ∩C l(V ) ⊆ [∪x∈K0 f (bC l(Ux))] ∩ C l(V ) ⊆ ∪x∈K0 [f (bC l(Ux)) ∩ C l(Vx)] = φ. Therefore, we have y ∈ C lθ (f (K )) and hence

f (K ) is θ-closed in Y . D

IJSER

Theorem 3.17. Let X be a submaximal ex- termally disconnected space. If a function

f : X → Y has a b-θ-closed graph, then

f −1(K ) is θ-closed in X for each subset K

which is quasi-H -closed relative to Y .

Proof. Let K be a quasi-H -closed set of Y

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∗ Department of Mathematics, Rada’a College of Education and Science, Albida, Yemen, ∗

∗∗ Umm Al-Qura University, AL-Qunfudhah University college, Mathematics Department, AL-Qunfudhah, P.O. Box(1109), Zip code, 21912,KSA

E-mail address : ∗ hakim −albdoie@yahoo.com,

E-mail address : ∗∗ alitaani@yahoo.com,

IJSER