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

ISSN 2229-5518

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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 l*θ (*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 l*θ (*A*)) *⊆ C l*δ (*f *(*A*)) *for each sub-*

set A of X ;

1(*B*)) *⊆ f*

1(*C l*δ (*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 l*θ (*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 l*δ (*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 l*θ (*f *−1(*V *) *⊆ f *−1(*C l*(*V *)) *for each*

β-open set V of Y ;

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

b-open set V of Y ;

(4) *bC l*θ (*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 l*θ (*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 l*δ (*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 l*θ (*A*)) *⊆ C l*δ (*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 l*θ (*f *−1(*B*))) *⊆ C l*δ (*B*) and

shows that *f*

−1(*F *) is *b*-*θ*-closed in *X *. There-

hence *bC l*θ (*f *−1(*B*)) *⊆ f *−1(*C l*δ (*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 l*θ (*f *−1(*G*)) *⊆ f *−1(*C l*δ (*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 .

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 ×

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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

0

(*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*

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U0 ∈ (BO X)0,axnd b0C( lX ) U0

⊆ (bC )l .U0

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Therefore, we obtain (*f |X*0)(*bC l*X0 (*U*0)) =

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*- *T*2 ( 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 *rT*0 [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) ∈

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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.

Therefore, *X *is *b*-*T*2.

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*(*U*1)) *∩ g*(*bC l*(*U*2)) = *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 ),

D

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 .

7

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 *{V*α : *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}).

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**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 *K*0 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

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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of *K *and there exists a finite subset *K*0 of *K*

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open in *X *and *bC l*(*U*y ) = *C l*(*U *). Set *U *=

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8

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[9] N. Levine, Semi-open sets and semi-continuity in topological spaces, *Amer Math. Monthly ***70 **(1963), 36-41.

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[16] J.H.Park, Strongly *θ*-*b*-continuous functions

Acta Math Hungar. **110**(4)(2006), 347-359

[17] J. Porter and J. Thomas, On *H *-closed and min- imal Hausdorff spaces, Trans. Amer. Matm. Soc.

[18] N. V. Veli´cko, *H *-closed topologicl spaces, Amer. math. Soc. Transl., 78(2)(1968), 103-118.

∗ 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,

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